I haven't uses patches myself, but heard of other guys using it. It seems that they are one of the lesser used ways to obtain testosterone. The main downside of patches were the fact they got very itchy after a while. Of course this may differ from person to person, but having a patch on your skin 24/7 may start to irritate you. I don't know how well they work dosis wise.
I did use androgel for 2 months, and was happy about it. My blood levels were checked and my T levels were in the high range. So it definately works. When I asked the doctor about the dangers of transferring it, he said it was very minimum. Just apply it, let it dry, stay clear of having that area touched by anyone for half an hour and you're good to go. Of course very close skin to skin contact (think of hugging or more) may be slightly dangerous.
It is clear that your doctor does not have a lot of experience with trans people, so looking for an endo who has might be a wise thing to do.
I can't say much about slow or fast changes because I only used androgel for 2 months.
Also, comparing dosages of different application technigues doesn't do much good, I think. Patches may have a completely different form of T that requires a different dosage than androgel.
Congrats on starting T!