He did tell you about his illness though, enough that you know he's in pain and needs a cane sometimes. I'm not sure why you need more than that from him? It's his job to monitor his fitness to practice, not yours. If he turns up for session I'd assume he can handle whatever I have to bring - good, bad or ugly - and I'd address anything in his response I wasn't happy with. He can be entirely authentic while keeping a boundary about sharing his health concerns, it's not really any of your business what the condition is, if there's anything affecting your sessions - health included, I'd just deal with it as it came up.

He should remember your name, not least because I assume it's written in his appointment book, and the dbt thing wouldn't be ok for me at all, but I'd have said that right at the start rather than being non-committal or partly agreeable, because group work is a huge deal for me. The authentic/not perfect thing I'd be less bothered by, but I wouldn't let him off the hook with it - I'd be asking whether that meant he expected me to live with him not doing x, y, or z and that if I'm being genuine, his lapses aren't ok with me and I need him to tighten up a bit.

Basically I think there's joint responsibility in setting up a new working relationship and being clear about what's ok or not for you is part of it. For example, someone being consistently late, or using their phone in session is never going to be ok for me no matter how "authentic" it feels for the T to always be contactable or march to the beat of their own drum. My job to communicate that clearly and then decide whether I can work with them or not depending on their response.