A lot of questions there

To answer the basic ones quickly and get them out of the way.
There are options other than shots, but shots seem to commonly be the route to get fastest results. There are pills, patches, gels... I think that's all but honestly I don't always keep up the best on these things since I'm old school and do shots lol.
Shots can be done in tons of places and will depend on if your doctor wants to do intramuscular which is the most common. I do intramuscular and can choose between my butt, thighs and arms (depending on the amount of muscle you have in your arm, I still need to do more lifting to build my biceps before I could go that route).
As far as how to go about getting on T, that depends on a lot and takes a lot more discussion. Have you talked to the counselor about rather or not she is even able to write notes? Depending on her degree and her job she may not actually be qualified to do so. You may need to see a regular therapist or even a specialist who works with people who are trans or have gender related issues. Where I live (Omaha, NE) there are literally 3 or 4 therapist who will write notes for sure after they see you for awhile (each one is different in their personal standards of time period). Sometimes I've heard of psychiatrists writing them as well. You may want to contact your doctor and ask their advice. There may be specific types of providers you would need to use in your area, for example here my endo does my HRT but my regular doctor (internal medicine specialist) will not provide that service. It varies from place to place and doctor to doctor it seems.
If you are getting no where with the counselor though I'd definitely say ask your doctor for some advice on who to turn to where you live.