The only real precautions they need are to keep everyone at least six feet apart and wear masks. And since Trump is the one with the virus, it'd be pretty simple to keep people away from him. Don't see why it would be more difficult than in any other place. Seems like it would be easier than in a hospital if anything, although it's a military hospital, and they probably have some special presidential suite or something.
I really don't see any way this helps Trump. For many voters, the #1 issue is controlling Covid. And, despite whatever sympathy he may get, it's not making him look like he's competent in handling it. Obviously, they'll try to spin it to make him look better, but I don't think it will work. Especially since Trump has been really working the "sleepy joe" "low energy" "diminished" angle.
Plus, Biden also has a campaign, and they will also spin, plus he's actually able to talk and reach out to people. I don't think he'll play the "Trump is weak" angle, but he'll, rightly, point out the serious threat Covid poses, that nobody is safe, and that they need leadership that is going to effectively handle the virus. Trump's handling of the Coronavirus was already his achilles heel. He doesn't need that to be the #1 issue.
As for how this works, it goes by the 25th Amendment. Trump wouldn't have to step down, but he can temporarily give control to Mike Pence, and then can resume power once he is capable. They did that when Regan had to go under anesthesia for some procedures. Pence and the cabinet can also, if Trump is not willing or able to temporarily relinquish power, declare him incapacitated and assume power. I believe a supermajority of both houses can do the same thing as well.
If Trump is incapacitated, Pence becomes VP. He would have to nominate a VP that would be approved by both houses, if it's expected to last for a while. If Pence becomes incapacitated, then Nancy Pelosi would become acting president. If she is incapacitated, I think it's the president protempure of the Senate.
The more covid could potentially spread throughout the WH, the worse it would look for Trumps campaign. Taking extra precautions to make sure it doesn't spread, wouldn't be the worst idea politically. If they can keep Melania more stationary, then I can see why having her there wouldn't be that large of a concern. Keeping Don stationary seems quite unlikely, so taking him to a hospital where he'll get separated with special treatment doesn't seem ridiculous. Plus he likely wants to get well asap, and going to a hospital where his risk specifically, will be super low, isn't another terrible idea. Could potentially cause further illness, but likely not
Trump is pushing to reopen things. A big reason why not to is because of people being scared of the illness. If Trump can survive it with ease, then it will likely make people feel more comfortable going out. Users here are always saying the Prez has major influence over the people in plenty of ways. Therefore much less reason to keep things locked down while reacting so harshly every time a spike of any sort occurs. The Dems are about lockdown, Trump is about reopening. If Trump doesn't get too ill and makes a reasonable recovery, he'll no doubt point out how he fought off weak covid and still has more energy than Joe. That's almost a given.
The people who are mailing in ballots have decided already, period. The undecided won't mail until they decide or run out of time. Trump doesn't seem to want voters voting any earlier than they have to. This will likely make undecided voters wait. It can also give Trump a chance to appeal to them, especially when it comes to dealing with covid, assuming it goes well for him.
If he makes a fuss about handing things over to Pence that'll surely hurt him, but I don't think he will if he looks to become quite ill. If covid goes poorly for Trump, like he's out of commission for more than 2 weeks, it's going to be an accent of a lifetime if he can get re-eleceted.