Now, I like Sailor Stars, but I must say this is one of the most frustrating and bafflesome plot points in Sailor Stars. (Although I suppose it's at least still better than if they'd completely contradicted the anime version and not mentioned Professor Tomoe... it's kind of amazing they brought him in at all given the elimination of most non-Sailor Senshi characters that season.)
All the information we actually have is (unless there's some guidebook or interview that explains it... maybe TuxedoUnmasked can solve this mystery with obscure research?), Professor Tomoe survived the destruction of Mugen Academy but gained some degree of amnesia about the events (and seems to have stopped wearing glasses; perhaps he has a subconscious phobia keeping him from getting new ones?). He was wheelchair bound at the end of S but this is no longer the case in Stars. Setsuna shows up and says something vague to him, and then... we have no idea what happened to him next.
What we do see is Hotaru living with the three Outers somewhere (I don't think it was Tomoe's house? I could check the set design since we did see both, but, I'm pretty sure it wasn't) as she matures, having various visions, including memories of her dad dive-tackling her out of the way of an explosion (I'm not 100% sure when this is meant to take place since in the flashback in S it appeared that he wasn't there to protect her in the lab explosion, but, it's a cool scene regardless). At the same time, she's started saying Haruka-papa and Michiru-mama, so... it almost seems like they just adopted her. Every time after that, she seems to appear alongside them, well after the Nehelennia arc is over, and in the finale she's hanging out with them by the car talking about the shooting star.
So what happened? I can imagine all sorts of guesses, but they are only guesses:
- 1. - Maybe Sailor Pluto thought it was too dangerous to continue leaving Hotaru in the Professor's care until they figured out what they were up against, in case he got possessed again. She might've just abducted the baby with no explanation, or she might've told him something, but really, how often does Sailor Pluto really explain things to people? I'm still not even sure if Haruka and Michiru knew Usagi was Princess Serenity / to become Neo Queen Serenity during S, since they seem to respect "Small Lady" but have zero respect for Usagi until after the Mugen Academy ordeal is over (and their respect for her decisions is still horribly limited as we see when they attack the Starlights... honestly, this is the kind of stuff that causes civil wars in monarchies, they're basically treasonous, but enough about that).
- 2. - Maybe the Professor lives with them / they live with the Professor, but he's simply not shown. (This seems unlikely since he doesn't show up no matter how distressed she is, but it's not specifically ruled out... I dunno, maybe he had a job to go to or something??)
- 3. - Maybe the Professor objected to Sailor Pluto taking Hotaru, and she killed him! (Doubtful, but would we really know the difference?)
- 4. - Maybe Hotaru spent the Nehelennia arc living with the outers, then moved back in with her father during the Shadow Galactica arc and just happens to never mention it and has to meet up with the other Outers separately.
- 5. - Maybe the Professor got sent on some sort of mystery errand by Sailor Pluto that kept him busy/absent for the duration of Sailor Stars?! (Pardon me, I smell my unfinished fanfiction burning in the oven)
Also mysterious is that Pluto seems totally caught off-guard by meeting the Sailor Starlights, like she doesn't know about them. So I guess there's some sort of timeline weirdness connected to their presence? She also doesn't seem to know what's going on with ChibiChibi, which suggests Galaxia's entire history may be beyond the scope of Pluto's 30th century memories... so spacetime sounds like a confusing mess.
As far as Setsuna and the Professor taking care of Hotaru together, that is an interesting idea. I read part of a long fanfiction (which I want to finish eventually since it's interesting, but it's a bit garrulous (like me) in parts/very high word count) called Sailor Moon Z, where they became a couple.