Do colleges know where else you've applied?

And if they do know where you’ve applied, will they know of your correspondences and interactions with other schools?

No

they might ask. I had my UChicago interviewer ask where else I was applying but that was the only school out of the four that I have interviewed with.

If you are required to fill out the CSS Profile for financial aid, then I think they can.
I know in the past that they could through FAFSA too, but I’m not sure if that is the case this year.

There have been news articles about some schools taking this info into consideration.

See this: http://talk.collegeconfidential.com/financial-aid-scholarships/444756-fafsa-css-can-schools-see-what-other-schools-youre-sending-it-to-p1.html