From what you've said, i didn't hear that she was ridiculing you when she suggested changing your looks (obviously she didn't mean changing your skin colour imo). I saw it she was suggesting looking at the problem from a different angle, from various different angles; circling the problem to see if anything fitted. It is possible to change our 'looks' in a variety of ways: dress, haircut/style, contacts/glasses etc. That's how I read it, that she was trying different angles/solutions.
I also hear 'but he's your dad!' as a reference to the universal pull of a blood parent, whether we want it or not. Or are able to acknowledge or access it: it is there. Imo this is what she was referring to.
As for being sectioned, it is nigh impossible to get sectioned so you don't need to worry about that. An awful awful lot of people need to be sectioned, or at least to be offered residential mh care, but the funding simply isn't there except for at the severe end of things - and you don't sound as if you're there. I assume you're still working eg ie functioning, even though you may be feeling very unsettled at the moment.
I'm so sorry to hear you are suffering so much. Please be assured these rocky times pass, they really do. I know to let them fling me about, even tho I don't know where I'm going to end up: bracing can cause more damage?
I hope you get good support from your gp. In the meantime do contact the Samaritans, they are very good when things are bleak - though if you feel the person you are talking to isn't doing it for you just cut the call (you don't even need to sign off, just replace the handset) and dial again to get some one else.
I'm glad you posted