The UK did NOT "vote for Brexit". It was a referendum. A poll, if that's easier for you to understand.
Semantics. There was a referendum on whether or not the UK would leave. Whether or not the result is binding is immaterial in this context; more people voted for leave than did remain. (I didn't, as it happens, so don't bother lumping me in with the "Brexit means Brexit" crowd.)
There is no law or legal obligation to actually go out and withdraw from the EU as a result.
Technically true, but actually going against the result would be political suicide even if the whips could manage it.
The current government, a right-wing government (Trump), agreed with the outcome of the poll...
No more right-wing than the last one*, and fuck all to do with Trump. It's debatable whether or not they agree either; the government's official position before the vote was to remain in the EU. We have disingenuous frauds like Farage, duffers in the Tory back benches, a lacklustre remain campaign and unprincipled media outlets to blame for the leave vote.
...and hopes to stay in power by implementing the result preferred by 51.9 percent of the people.
Well, duh. Of course they hope to stay in power, have you ever seen a politician who didn't?! The result was actually 37.4% of the electorate voting for leave, vs. 34.7% voting remain. Only 26.7% of "the people" voted to leave the EU. Sorry if that sounds petty but I cringe whenever I hear a politician waxing lyrical about their so-called mandate from the people when they have don't even have a majority of the electorate behind them.
They have no legal obligation to do this.
No, but as said earlier they'd be insane not to follow through. The best they could hope for is a second referendum on whether to accept the terrible deal we'll end up with.
They have already lost one legal challenge, which they are appealing and will likely lose a second time.
You're talking about the high court ruling, which was that the government cannot trigger Article 50 to begin the process of leaving without a parliamentary vote first. On face value one might think that they really want to leave, but the government is likely just worried that either MPs (on both sides) will not vote in favour of leaving or that somehow the UK will tip its hand when it comes to treaty negotiations, so they want to be able to start the leaving process without public scrutiny. In reality I don't think the government wants to leave the EU any more than I do but that's Cameron's fault for not standing up to the old guard in the Tory party and giving us a referendum in the first place.
*So far, anyway. I'm of the opinion that home secretaries make bad PMs: their job is to maintain order and they tend not to be too concerned about liberty when they go about it. When they have that mindset and actually have the power to follow through on it we end up with things like the snooper's charter.