1. Will I get boarding passes for both flights, JFK to LHR on AA and then LHR to MAD on IB, when I check in at JFK?
YES if both on the same ticket
2. Will both boarding passes show my AA Platinum status?
YES
3. Since I will not have my AA AA Platinum by the time I travel, will my access to BA LHR T5 lounge be denied? If so, any suggestions for how I could provide the proof of my status?
Usually, they just scan your boarding pass and you will be allowed in.
Sometimes they do ask for your card. You can print out the AA webpage with your name and status.
Or you can call AA, explain the situation and have an email or fax sent to you as proof of status.
I can't remember if the OW status
Sapphire shows on AA boarding passes in addition to
"PLATINUM". Just in case you encounter a newbie (I doubt you will need it), also print out a page where it shows that AA PLAT is equivalent to OW Sapphire like this one:
http://www.oneworld.com/ffp/my-oneworld-tier-status/
You will be more than prepared with all these
Welcome to AA! and enjoy the Galleries at T5!