What do you call:
The headband with the little reflecting circle doctors used to wear?
No idea. I assumed it was just a head torch, and they were used because handheld torches were too cumbersome, and large overhead lights weren't powerful enough. Not used any more in the developed world.
The thing that doctors tap your knee with? It has a little triangular wedge at its end...
A tendon hammer or reflex hammer. Can be any shape. Circular ones hurt less.
The satchel that the home-visiting doctors used to carry?
You mean a Doctor's Bag? I've never come across a special name for them. Bag of tricks maybe?
The machine that shows the contractions when you're giving birth?
A CTG (cardiotocograph) machine. Measures the baby's heart (cardio) and the mother's contractions (toco).
And what do you call the paper that the above machine spits out?
A cardiotocogram. As to the name for just the paper, ask the manufacturers.
Is the throat swab (the one on a long stick that looks like an elongated Q-Tip) simply called a throat swab?
Yep. Maybe cotton-tipped swab is the more general term, but if you ask for a throat swab, you'll get the same thing.
And what about that thing that takes your blood pressure--that band that feels like a boa constrictor?
Sphygmomanometer, or sphygmo. Or BP machine.
And the heavy blanket put over your chest when the dentist takes xray pictures?
Lead blanket sounds good to me. The things you wear are lead vests or lead aprons.
And, finally (this is the most important), when you have nearly microscopic veins, as I do, and you want the nurse taking blood to use the very smallest needle, is there a particular type of needle I could request--say, one for new-born infants?
As already mentioned, smallest is not necessarily best. You can try asking for a 22-gauge needle. (I think a 16 or 18 is normal, but I could be way off).
Many thanks to anyone who can provide any of the above terms.
Note the edit. Got me numbers a-muddled.