13.56MHz RFID Antenna design
Unfortunately, it is not easy to design a good 13.56 MHz antenna. In general, a good antenna is at least 1/4 of the operating wavelength (of ~22m), so any practical 13.56 MHz antenna would be a bad antenna. 

The main problem with electrically small antennas is that radiation resistance is tiny (less than 1 Ohm), making it almost impossible to match it to the 50 Ohm. One technique for matching would be to introduce a small series resistance (e.g. 2 Ohm) and then match to it with capacitors and inductors. However, this approach is difficult without access to a vector network analyzer. 

I noticed that the module has a UFL connector. One way to improve the antenna would be to plug in another antenna you buy off the shelf and disable the current one (will probably need to cut traces of the old antenna)

To answer your questions:
"How important is material choice (wire/tape/tube)?" - not important at 13.56 (as long as it is metallic)

"Do I need an additional power supply" - not if you are using the same chip

"Can I build a square antenna into the flashing around the card window" - would need more info on this.