Re: Before 7:34pm.
Clara definitely understands the need to be able to figure things out on your own for sure. So she won't be offended at all.
If she introduces herself as SIR Integra, Clara will, of course, do the formal British thing because that's how they do. Shake hands with Alucard. Etc, etc. Clara will tell her that she's been bouncing around with most of these people for the better part of 2.5 years and that yeah, mogwai presents and costume things are pretty normal. There have been times people changed genders and bodies. And another time they went to a small town and had completely different lives and didn't remember their own.