There are several online scheduling tools out there that you can try. In my opinion, the best one you can use that you can integrate into HubSpot is Chili Piper.
It is an advanced scheduling platform that allows both your staff and clients to schedule appointments based on their availability. Chili Piper is also useful in qualifying and routing leads, scheduling meetings from forms, and tracking engagements.
It is easy to integrate Chili Piper to Hubspot, and yes, I believe Chili Piper supports those things you mentioned. They also provide several solutions to different use-case problems. Just check their website or contact their customer support. They are very responsive!