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Can I replace my late 2013 MacBook Pro logic board with any A1502 logic board. Does it have to have the same amount of RAM and the same CPU as the old one?

You need to stick within the same sub group MacBookPro11,1 logic boards. So as an example you have a 13” MacBook Pro (Late 2013) 2.4 GHz Logic Board with only 4 GB RAM you could upgrade to a 13” MacBook Pro 13" Retina 2.6 GHz Logic Board with 16 GB of RAM Here’s the guide MacBook Pro 13" Retina Display Late 2013 Logic Board Replacement