Will I be held responsible for working with a supervisor accused of sexual misconduct?

The matter is best left to the official circles. Do not pry, do not gossip. As for being tainted, you are just a student and dependant, and if you neither enabled nor profited from the prof's (alleged, we don't know) misconduct, you should be fine. But if you start poking your nose into matters yourself, you'll get into a lot of trouble, from being seen as a nuisance to the officials, to getting the prof upset, to violating privacy laws, to unwittingly becoming a witness.

A career-limiting move.

Asking the Director of Graduate Studies (or equivalent) about whether there was an official resolution to the investigation is certainly appropriate, though be prepared that there’s a good chance they won’t be able to tell you anything. I would clearly stick to asking about official resolutions rather than their opinion on what happened. For one thing, it would not be unusual for a sanction from the university short of dismissal to include a ban on advising students! If so that’s certainly information you’d want to know sooner rather than later! I’d start with the DGS, rather than the chair, because issues around students finding advisors is more in their purview.

As for what you should do moving forward that is really up to you and your conscience. Personally I’m not comfortable collaborating with someone who I could not comfortably recommend as a supervisor to women students down the road. But I doubt that it would seriously harm your reputation (unless this professor gets fired for future behavior in which case you may be in a tricky spot), people understand that the advisor and student are different people.

There are a few aspects to this.

First, as Captain Emacs says, any allegations or hearsay that you may have encountered are just that - you have no formal standing to enquire (and certainly not to confront your advisor about this). Doing so will be a professional mistake.

Second, there is your perception. If you feel that having an advisor who was involved in unethical behavior is something you cannot do, then it is best to cut things off on your first year and find another advisor. Advisor/advisee relationships are more than just professional; you need to get along.

Third, there is your perception of university policies. Some universities are notoriously bad at handling sexual misconduct cases. If you believe that the university is not creating a safe environment for you or your colleagues, then this is something to think about.

Finally, there is the research community. From what it sounds like, you work in a relatively small subfield. Word gets around (perhaps less quickly nowadays with less in-person communication but still), and your name might come up, and this might be something you'll be asked about or have to deal with. You need to think about how to handle these potential interactions, and distance yourself from that incident.