How do I use RFID-RC522 with an Arduino?

Sorry, but I must disagree with your 'NXP datasheet is incomprehensible'. From the FIRST PAGE of the MFRC522 datasheet:

'Remark: The MFRC522 supports all variants of the MIFARE Mini, MIFARE 1K, MIFARE 4K, MIFARE Ultralight, MIFARE DESFire EV1 and MIFARE Plus RF identification protocols'

Here it says the MFRC522 covers just a part of ISO/IEC 14443. The following are missing fron the NXP RC522:

*MIFARE DESFire EV1 (includes AES encryption)

MIFARE DESFire EV2 (includes MIsmartApp, Transaction MAC, Unlimited Applications MIFARE Plus drop-in replacement for MIFARE Classic with certified security level (AES 128 based))

MIFARE SAM AV2 (secure access module that provides the secure storage of cryptographic keys and cryptographic functions)*

These are probably the ones used in bank cards. Again, from the FIRST PAGE of the MFRC500 datasheet:

'All protocol layers of the ISO/IEC 14443 A are supported.'

You'll have to plow through all the different MIFARE / ISO/IEC 14443 specs to find out how many ID bytes there are for the different types (I suspect diferent types return a different quantity of bytes).

So, basically, you're screwed. Spring for the expensive MFRC500-based reader. I assume that with the higher cost you also get a high-class API, documentation and examples or even (gasp!) tech support.

Cheers