This sentence really should be 见我的医生/to meet my doctor, or just 看医生/to see a doctor.

We don't normally insert 我的 into 看医生. 看我的医生 is not impossible though to stress this is my own business to see a doctor (a more advanced expression).
e.g. 我一会儿看我的医生，你就别管。
I will go my own way to see a doctor a bit later. That's not your business.
(Note: I modified this part 1 day later to remove a misleading generalization. Anyway, there is a problem to express in a quasi-possesive pattern when the relationship is originated from the action itself.)

Beware of different uses of 看:

• 我去动物园看猴子/I go to the zoo to see monkeys
• 我看小孩/I watch the children (take care of them)
• 我看他不会来/I think he will not come
• 我看人不会错/I never misjudge a person

U mom lol

I haven't heard about 一下儿. You probably are referring to 一下子？It means "instantly", "right the way".

It would be acceptable, provided a specific context. If I omit the subject "I" I would naturally also omit the "my" before "doctor". So if I keep the "my", most likely the implied subject is not "I", but someone else.

If we want to insert 会, we put it after the time factor and before the verb.
我一会儿会见我的医生
一会儿我会见我的医生
are the possible orders.

Putting it at the end changes the meaning to "for a while" rather than "in a while", as in "I will see my doctor for a while".

As it is a kind of time duration it can also be put at the end of a sentence. However, it is a short while only and hence we tend to not using it with an action that we do not have control.