Beid, I've paraphrased your questions. If I misinterpreted any of them let me know and I'll have another crack at it.
Oh, and another "wall of text" warning

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Also note that I am a lot less confident trying to answer these questions than I am in dealing with other aspects of finance and economics. This is a lot more speculative, and will be a lot more open to criticism.
I'll generalise in the discussion to come, but I think that, more or less, the thrust of what I have to say is probably about right. Just don't rely on it (and for certain don't make any decisions based on it!).
1. How far could this contagion, caused by the failure of the "Western" financial system go? Use as examples Russia, China, Singapore, Japan, Brazil, Jamaica and Malta.
The world is a much more interconnected place than it used to be, even ten or twenty years ago. There really is a "global economy". But it isn't equally interconnected. A place like Bhutan gets a few tourists a year, a place like the US is almost completely open to the vagaries of trade, globalisation and the markets...... This has brought huge benefits, and huge strides in standards of living for many countries that were, a few years ago, horrifically poor. But it has not been without cost, and there are still far to many people living wretched lives.
Benefits include much cheaper goods, increasing standards of living as a result. It's contributed to lower inflation, which results in lower interest rates (which, as long as things don't spiral out of control like they did in the credit crunch, is a good thing). We've also seen massive improvements in the standards of living in many poor countries, from Ireland to China. Twenty five years ago I was a child visiting my grandparents in Ireland. Significant swathes of the population still lived in poverty, for instance with no central heating, few televisions, poor diets, no jobs and no inside lavatory. Even after the most recent unfolding of events standards of living in Ireland are immeasurably higher than they would otherwise have been without globalisation.
Examples of costs are the wrenching dislocations experienced as industries and their staff are "offshored" (ignoring Canute-like Sarkozy examples, forbidding companies to relocate factories to lower cost locations - the results are inevitable). We have much higher consumption of everything, that has its costs, from cheaper food (more obese people), cheaper booze (more people with drink problems), more pollution as all these goods are shunted about the place, forced resettlement of people (eg in China), enormous disparity in wealth (think Russia more than the US or the west generaly), social unrest, environmental damage.... the list goes on.
In spite of that, I think that the balance is broadly positive.
So, the answer to your question is, "it depends". It depends on how enmeshed in the global economy a country is. It also depends on factors outside the "realm" of the global economy. The following will, absolutely, not be exhaustive, and in some cases may be entirely innaccurate. Let's take each country you mention in turn:-
Russia - Relatively integrated into the global economy but, and it's a very large but, it is not a democracy and not a market based economy. Russia is basically a kleptocracy, with a minimal few people at the top controlling it's economy. It may not be communist anymore, but all large investment decisions are at least guided by the Kremlin. It may export large amounts of natural resources (earning it US dollars), but what it exports, to whom it exports, and the price of those exports, is very much seen as a tool of state.
It is relatively insulated from the effects of the credit crunch, partly because the West (ie the global economy as it was then) has bitter memories of its default at the end of the 90's (meaning less foreign investment), and partly because it is commodity rich.
It is also dying. Its population, ravaged by poor health (drink, drugs, STDs) is shrinking every year, its infrastructure is crumbling, its politics massively unbalanced, and, because of the centralised control, it makes questionable economic decisions.
If I was Putin I'd be a lot more worried about the last paragraph above than I would be about the impact of the credit crunch.
China - Very integrated with the global economy. But in a very strange way. There are amazing stresses building up in China, both economically and politically, and the leadership really does have to walk a tightrope.
Its basic problem is twofold. Firstly the Communist Party must (in its eyes) maintain control. Secondly, in order to keep its population quiescent enough, it believes it needs to manage at least an 8% year on year growth rate. To achieve the latter it keeps its currency cheap (it is basically pegged against the US Dollar), to keep its exports cheap.
But what does it do with all the US Dollars it earns? It soaks them up (leading to inflation), and it buys commodities (leading to increased commodity prices for all other countries), and it "saves", by investing in foreign assets. Foreign assets like US Treasury Bills. What impact does this have? It keeps money cheap for the US, encouraging the US to borrow, further exacerbating global financial imbalance.
What it should do is allow its currency to float. This would result in a rise in its currency against the US Dollar. This would cause its exports to increase in cost (and decrease the cost of its imports) which would go some way to redressing the imbalances building up globally. To achieve its growth targets it needs to turn inwards and focus on domestic growth, whether that be making consumer goods more available to its population, or improving its infrastructure and services.
But this requires the Communist Party to relax its control.
So I have some sympathy with American politicians bashing China about its economic policies. I also have some sympathy with a derivatives trader I was talking to in Hong Kong back back a few years. He said to me something to the effect of "Christ, all these westerners bitching about China are talking rubbish. If they were in charge of a restless, growing, poor population of 1.5 billion, would they take the lid off? Don't be stupid.....".
Singapore - very open to the global economy, but also utterly dependent on the trade the drives the global economy. So in essence very vulnerable to the crazy whirlwind we're all experiencing at the moment. Except that is for fate. It's the largest shipping port in the world. And it's sat right on the pinch point between China (and Japan) and most of the rest of the world (excluding the US West Coast and Australasia).
As long as China continues to trade Singapore will be just fine.
Japan - Japan is an interesting one. Financially speaking it's very open to the ebb and flow of savings and borrowings. The problem is that foreign investment, seeking control (or even starting up new companies) faces huge cultural, bureaucratic and market based obstacles. There's no vigor in Japan (added to which is its rapidly aging population). And there's no spark to catalise that vigor. It's living on the savings it earnt during its huge post-war industrial expansion. They are running out.
It also lives in (and sees itself as an important player) in a volatile part of the world - a rising, expansionist China, a nuclear armed Korea that is fond of confrontation, an ailing America on which it relies for security....
Interesting times ahead for Japan, it will take political vision and courage for it to retain its global importance (and its standard of living).
Brazil - Brazil is very well positioned. It didn't take a hit from the credit crunch. It has a young a growing, increasingly well educated population, it is the regional power in South America. It has a strong commodity export position that sees increasing trade with China year on year. It's also introduced some (gentle, unlike China) capital controls to moderate the potentially extreme effects of foreign capital flows (think back to the Asian financial crisis - a wall of hot foreign money invested seeking high returns because of the "Asian growth miracle" story, all of which dried up overnight when the economies went "pop").
Jamaica - a relatively poor country, neighbour to the US, little industry, very reliant on tourism and remitances from Jamaicans living abroad (not sure about this last point, but I suspect it might be). It hasn't got far to "crash", because it was never very high on the hog to start with, but it will see lower growth and limited, if any, improvements in standards of living over the next few years. Tourism will be down. Remitances will be down.
Malta - I honestly haven't a clue. Other than knowing that it is a small island in the Med I know nothing about it. I suspect it will be even more insulated than Jamaica - it's richer, but has less dependent integration with the global economy precisely because it is so small. I'm guessing.....
So, to sum up my answer to question one, it depends

. You're probably beginning to get a feel now for why economists struggle to predict everything perfectly!
2. Of those countries not affected in the same way as the US and European countries (aka being "bought off by the IMF"), why are they not affected.
See above. Because they are different. Openness to the global economy is part of it, as is the "sophistication" of their banking systems, but there are many, many other factors in play.
Though I'm not sure I get your "bought off by the IMF" comment. The IMF is a "lender of last resort for countries". It is supported by many countries, largely in proportion to the size of their economies. It learnt its lessons in the aftermath to the Asian financial crisis. Back then it waited for things to go bad before stepping in to try and help clean up. Now it gets in earlier, and tries to forestall the worst of the problems. Let's see what the score is in five years time.
As for "buying countries off", it doesn't. It says "ok, we'll lend you this money you need, but before we do so we need confidence that we will get it back. The things that got you into this mess in the first place were x, y and z. Change them".
Say you spent $10,000 on "wine, women and song". Then say you borrowed another $5,000 and spent it the same way. Somebody lent you that money because you had a good job and they were confident that they would get it back. Only you lost your job because you'd partied way too hard. Now you return to your friend, your family, or your bank and say "Look, I really need help now, can you lend me another $10,000 to help me get back on my feet"?
A few conditions aren't unreasonable are they? Starting with asking you to stop partying.....
3. Why did the West not pay attention to the "financial models" of those countries not affected, and seek to emulate them?
Again, see the answer to 1. above. Countries are different. They operate in different ways, for different reasons, and are exposed to different opportunities and different pressures. The main reason that Italy didn't suffer directly during the credit crunch was because their banks weren't "sophisticated enough" to get swept up in the sub-prime / CDO crazyness. That bit of "their model" worked for them (but it would not have worked for the US, the UK, Ireland, France, Germany or Japan). Equally Italy has its own challenges, low tax take, corruption, abysmal productivity, high debts, and politics that would be laughable if it wasn't so distressing.
Some of the stresses directly exposed by the credit crunch are now exposing the fault lines in some countries' ecomies. The first example was Greece. Ireland was more directly hit by the fallout, but it took longer to show through. Italy has its problems, and it needs to tackle them. Otherwise it will get hit.
Chancellor Merkel said a day or so ago something to the effect that "Finance has to learn that it is subordinate to politics". Only it isn't. It's orthogonal to politics. She can't command people to lend. And when she suggests that she can people get scared, and take their money away. Every time she has commented on the financial situation that European countries face, and the finance industry in general, in the last six months, European borrowing costs have gone up. When will she learn?
I can't find the comment but Bill Clinton once said something along the lines of "I thought I was the most powerful person in the world until I met the bond markets". Chancellor Merkel needs to come to the same conclusion, quickly. She's doing as much damage to Europe's prospects as Greece and Ireland are combined.
Otherwise the bond markets simply aren't going to be there, and Europe will be bust.
I can't breathe.
- George Floyd, 25th May 2020