There are many threads on this please use Google and the forum to search.

Clogged sun roof drains, dead batteries in remote, leaking antenna base, leaking door seals and list goes on and on. These are all items to check as your BCM may have gotten wet.
What does the BCM control??? A ton of stuff!

 

I would say neither of your issues have anything to do with your BCM. Your truck is 10 years old, the TPMS have a battery life of...... 10 years.
It is not unusual that your fobs lost contact with the truck, happens sometimes when the batteries are changed or get very low.

 

Agreed with the above posters, change your TPMS sensors or ignore the warning (mine have been on for 2 years, I check my tire pressure manually almost once a week because I air down so much for wheeling, but I really need to defeat the system because the warning is annoying).

Change your remote batteries and if you're still having issues, you might need to take it to a dealer to have the remotes reprogrammed. Needing to do this would be super rare, but if it doesn't work, then that could mean there is a BCM issue. But I'd replace the batteries before jumping to any conclusions.