This would be the first drug test in the history of drug tests that I support.

Honestly? I think a probation officer is likely to be picked for seniority in his case, for the purposes of making the new York justice system appear more, frankly, funded than it really is. I don't think the political pressure on that officer is likely to be the same as the political pressure on someone like a judge. Their pressure will be to make the department look good.
That will inevitably be interpreted as professional, courteous and unbiased.
I don't think he'll be drug tested, but I do think he'll have at least one off hours home visit that is coincidentally picked up by the media, and they'll find a reason to deny some request until additional concessions are made.
At some point someone will say that as far as they're concerned, the former president is just another individual under their supervision who's being treated just like any other. It's not true, but it requires some indignity to happen to be plausible, just because everyone is watching.

Enough so that no one says they're obviously just weak, since that makes them look bad, but nothing so much as to make people say "wow, they're actually awful, we need to fix the system".

Alternatively the system assigns someone at random who more than likely power tips a little and takes advantage of the opportunity to abuse a rich person just because they can. Probably not though, since that requires a different level of institutional incompetence.

Yes. The system is designed to allow the probation officer a lot of leeway in how they enforce things.
Usually the system is petty and difficult to work with, but it's not obligated to be.
They could say something like a biweekly phone call and a call before traveling out of state is all that's required.

Unlikely though, since blatant deference makes them look bad, and opens up whichever political appointee or elected official that sits above the probation system to an easily justifiable replacement.

I mean, probation is the "slap on the wrist" punishment.
There's an actual possibility of him going to prison. That's not a hypothetical, or even unrealistic. Cohen served prison time for this same act, and he pled guilty and agreed to testify.

Trump is unlikely to go to prison due to the politics of the situation, but it's not impossible. If he doesn't go to prison, the only other punishment is probation, which carries with it things like travel restrictions. Those aren't negotiable. They're unlikely to deny his requests, but he would actually have to make the request, and check in with his court appointed probation officer at least by phone on a regular basis, or that officer can just send him to jail for non-compliance. A home inspection is standard procedure, and the officer doesn't need anyone's permission to do it.

If the judge that Trump has continuously attacked decides that Trump, who has shown zero signs of remorse or contrition for his crimes, deserves leniency then I see no reason to expect that the terms of his leniency would be exceptionally lenient, to say nothing of "in open defiance of the minimums of what's required of probation".

IANAL, but I believe that is up to the judge's discretion, as is the sentence itself.