Can a parent and child share a psychiatrist or is that too weird?

Sharing not best. It is often best for the child psychiatrist to treat the child/adolescent alone so that the pt. can establish a rapport, trust and condifentiality with the doctor. Although the child psyhchiatrist can treat adults in order to maintain the relationship with the pt. The mother should have her own psycih. If the town is small & there is only 1 psych. then an exception may be made.
Yes. But it may influence the doctor's advice despite their best efforts to seperate the cases. This may be a good thing or a bad thing.
Yes, but. Some psychiatrists will choose not to, depending upon the situation. You should ask them before initiating treatment.