Can Thomas talk/send images/send notions into your character's mind? Yeah sure! Pretty sure he did stuff like this with Solas anyway, though since he's more real than spirit he may not be as good at it.
Can Thomas poke around in your character's head/explore beyond just communication? He probably wouldn't be able to since Cole isn't actually. human. I mean he can try but I'm not entirely sure he'd be able to make sense of anything before Adamant Fortress, when he becomes more real, and even then it's a toss up.
To what extent is it cool for Thomas to explore you dude's brainpan? Cole doesn't really have anything buried and like I said, I'm not sure he'd be able to make sense of much of anything before Adamant Fortress. He's welcome to try!
What kind of stuff would be cool to Thomas to dig up, if he spends more time snooping in your character's head? Well the whole "not human" bit would be pretty obvious, since he's still learning to think like them. He thinks of things as they might be in the Fade, so it'd probably translate into impressions and stuff, not concrete thoughts? But he can pick up anything that's on the wiki, really.
What stuff do you NOT want Thomas digging up in your character's head? We're good.
Do you want to do any kind of bleed over thoughts/feelings/memories if Thomas is flipping his shit while mentally connected? Cole actively goes looking for this kind of stuff, even if he isn't able to do it as well as before, so this is totally okay!
Contact? PM or add me on plurk or whatevs!
Anything else? idek