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I am confident that if you research this further you will be able to answer your own good questions but I offer this from my limited knowledge:
Using 13.560 MHz how do more than one RFID chip manage NOT to interfere with each other?
Each unit only responds when queried by a unique digital tag from the transmitter. Thus, all are silent unless spoken to.
How do the chips, those attached to a product or implanted in a pet, get their power?
They get their power from the RF from the query transmitter.
Quick, short, and possibly wrong answers but this is based upon what I have read. Let us know when you find out.