Okay now the claim that they prefer Trump makes more sense to me, because I know that he shafted both through sanctions (both state and persons).
The heaviest sanctions really date from Obama. The Trump administration has added a few more for various infractions, although Trump personally has an expressed a desire to end them. (We might note that Russia has been a significant source of money for Trump's to finance his businesses for years, and he was working hard at a project in Moscow before his presidency.) We can probably be sure that Russia will think it's done better with Trump than it would have done with Clinton.
My take on Trump is that he is a largely disinterested leader. He's completely self-absorbed, so all he really cares about is self-gain and how he appears to others: as a result he leaves governance to flunkies, and involves himself mostly only when he perceives benefit to himself. We can see evidence of this in his debates - he can't discuss anything in depth, and that's because he doesn't know any detail, and he doesn't know because he doesn't care about it and doesn't do it. So in terms of Russia the State Dept. will go do their thing, and although Trump will grumble and would probably like to end sanctions for the sake of his businesses, it would be far too costly for his image as president given his existing Russia woes politically to push it as policy.
What foreign leaders are looking at with Trump is his evident hostility to a lot of international norms. He's alienating allies, he's been negative about NATO, undermined the WTO, he supports the break-up of the EU: fracturing of the alliances and systems which the West dominated the postwar global scene. Iran, for instance, is a prime example. Rather than co-ordinate a joint response, Trump just wanted what he wanted and seriously split the US and EU to do it. In the long run, this loss of trust and co-ordination weakens the USA and EU and their allies - and other countries take advantage. So Iran might struggle under Trump's crosshairs for a few years, but letting him swing a wrecking ball at the USA's reputation is a benefit for decades to come.
yet NATO did fuck all to stop Russian aggression against Georgia in '08.
It had no obligation to, and it's hard to see what it could meaningfully have done anyway.