Doctor/Bloodwork:
It depends on the labs ordered, but it's generally 2-4 vials, not a lot of blood. I don't have a fear of needles, so I can't help you much there. It's highly unlikely that the person drawing your blood will nick an artery. Veins bleed much more slowly than arteries, so if you are worried about bleeding too much, you should be okay. And unless they are testing specifically for drugs, no. Even if they were testing for drugs, nicotine wouldn't be something that would show up.
Endo:
It's possible depending on your endo. If you have blood work beforehand and everything checks out, you might start T that day (in office or at home with a prescription).
Physical:
I can only speak from my experience, but my endo did a brief physical on me. Fully clothed, just the basics (heart, lungs, abdomen). Thankfully my endo is knowledgeable about trans people so she knew what my binder was for. No pelvic exam, but we did talk about periods and safe sex.