Sure if it's an emergency. If the 15 year old needs to see a doctor to save their life, of course they can. If it's not an emergency, then the 15 year old will probably need their parent's consent. Doctors charge for their time, and can the 15 year old who goes to the doctor pay the doctor's fees for seeing them?
It depends on what state you are in. Do a search on the internet regarding the above in your area/country.
no
Okay, this decision is actually more up to the parent than the state law. Yes, a 15 year old may be seen by a doctor alone, without a parent. But, until they are 18, it will still be up to their parents and/or guardians. Doctors usually prefer to see patients by themselves, to preserve "Doctor-Patient Confidentiality".
no
No
yea
No
Yes.
There are no legal barriers to that relationship.
I'm not sure if a 13 year old can actually buy diet pills, but I do know you should first ask a parent or a doctor before beginning the pills. Therefore, if a doctor or parent allows this, the doctor can give you a prescription or the parent will go with you to buy the diet pills.
The custodial parent.
No.
No.