Does your character have any immunity or defense against being empathically read? Nope, none whatsoever.
Is there any particular pain and/or memory you would like Cole to pick up on first or focus on? He tends to be the type that worries more about his friends than himself, so that would probably be stronger than any other anxiety, at least at first. But there's also the fact that he's been with the Doctor for about three years at this point, and while he loves the excitement of traveling through time and space, he's been away from his native Scotland for a very long time now and misses it more than he generally lets on. The fact that he's never managed to get back bothers him and being in this new place without any way out of it is likely going to make that worse, at least in the short term.
Is there anything else that can be found in their app or wiki pages that should remain off-limits? Nope. The one thing that will truly upset him - losing most of his memories of his travels with the Doctor - hasn't happened yet at the canon-point I took him from, so everything is pretty much fair game. (For reference, it's from near the end of 'The War Games' when he's wound up on Gallifrey, but before the Time Lords mess with his memories and send him back to Scotland.)