For antenna design notes, check posts in this section of the forum; http://rfidtoys.net/forum/forum_topics.asp?FID=21&title=howtos-and-other-projects

The book is now quite dated, and there are several Mifare RF readers available that you could attach your own antenna to. The trick will be to properly tune the capacitance of the antenna, but many of the OEM reader boards designed for use with an external antenna will already have these tuning capacitors on the board, and you'll just have to make sure you wind your antenna to the appropriate inductance.

I think the main issue with an RFID enhanced stethoscope is going to be the metal portion of the stem that the drums of the check piece and diaphragm attach to. If possible, you will want this to be made of plastic. If you can't find one with a plastic stem, you may try 3D printing one.

We obtained the RFID Toys book and a number of other materials. Since I'm not very affluent at physics, creating our own antenna is quite a challenge (although a physics teacher colleague is looking along and he might figure things out).

I have seen the SkyeTek's SkyeRead M1-mini in the book of course but it is quite expensive, not very available (especially the TTL variety) etc. We already have a bunch of readers and want to connect it (initially to an arduino or other uC that we are familiar with).

The diameter of a typical stethoscope is about 4.3 mm (1.7 inch). We might be able to create out own prototype PCB antenna by etching it, but we still hope to find out what our best options are for now.

(so any help or suggestions very much appreciated), thanks!

Most likely it works like this; where Mifare Ultralight cards have UID bits written and locked at the factory, they are likely selling special tags with OTP or one-time-programmable bits where the UID bits of a normal Ultralight tag are, and the programmer simply reads the UID bits from the source tag and writes them to the target tag. I'm pretty sure the magic all resides in the special blank cards they sell. You might even be able to program them with a normal RFID writer and just target those memory blocks, but that's just a guess.

From my understanding Ultralight (non c) uses a slightly different communications protocol than classic but for the most part is physically the same. For cloning classic cards I use a UID changeable card, does anyone know if that is the route I should take with Ultralight? I haven't even been able to read block information besides the UID on proxmark.

Any help would be appreciated!

Sure  they are - same seller that has been selling for years