Most of these questions are of course unanswerable. For one thing the government of the Ukraine will play a major role in determining the post-conflict landscape, and their positions are not yet clear. The US and the EU will have very substantial influence over the post-war Ukraine, which will be an economic dependency for years to come, but I personally think it would be a bad idea for the US/EU to degrade Ukrainian sovereignty by dictating policy. It will be a fairly delicate bit of balancing.
We don't know. Part of the problem will be differentiating between ethnic Russian interests and Russian national interests... for example, the Russian demand for autonomy with a veto over foreign policy decisions is clearly intended as a Russian level over possible NATO membership and other links tot he West, and is incompatible with Ukrainian sovereignty. At the least the ethnic Russia community could be offered recognition of their language as official, as the Quebecois got in Canada. It would help a great deal if the ethnic Russians can develop a moderate leadership that can articulate expectations and desires of the community without being controlled by Putin. Whether or not that is possible we do not yet know.
Ukraine will be effectively dependent in economic terms. The extent of the assistance they receive, and the conditions attached to that assistance, will have to be carefully worked out.
Ideally you'd send the Russians back to Russia and let Putin deal with them, and allow at least the rank and file of the local insurgents to stay without penalty. Of course that is hypothetical and the governments in question will have a lot to say about it. There may be some agitation among ethnic Russians for full scale relocation to Russia. There is some precedent for this: much of the ethnic Russian population of Kazakhstan has returned to Russia. Whether the Russians would be amenable, or how it could be done in a way that doesn't look like ethnic cleansing, is anyone's guess. My guess is that it will be handled badly and make a mess.
Not possible to know or control.
I think the risks to "the relationship" are in many ways overrated. The commercial interests on both sides are too strong to repress out of enduring pique, and I'd expect trade relations to be renewed pretty quickly if the conflict is resolved. If the resolution of the conflict is handled in a way intended to inflict outright defeat (or emasculation) of Russia, we can expect them to disrupt our strategic interests to the greatest extent of their ability. Of course that doesn't have to happen.
The problem here is that "Russian security interests in Europe" seem to require an allied or at least neutral Ukraine, and short of outright conquest that no longer appears to be achievable. How that sorts out is anybody's guess. The west cannot promise a neutral Ukraine, because that would intrude on the sovereign right of the Ukraine to choose its own alliances. If the Ukraine goes firmly pro-west, even without Crimea, the only ally the Russians have on their western border is Belarus, and that's a shaky ally at best. If Russia can't control the eventual transition out of Lukashenko's rule, they may be left with exactly the situation they want to avoid: the West on their doorstep with no buffer. What Russia will try to do about that is, of course, up to them.
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