"Info" (info) is short for information. It's a more informal way of saying "information."

Next Alex says that he is passing around a list with contact information (names, telephone numbers, and email addresses) for everyone.

"To pass around" means to give to people who are usually sitting in a group; they might even be sitting in a circle, but not necessarily.

"To pass something around" means to give it to one person, and then give it to the next person, and so forth, in a group.

Alex is going to pass around something that has contact information for everyone.

He asks people to check off their names once they have doublechecked that the information is correct.

"To check (check) off something" means to put a check symbol next to something on a list - a little mark on the paper.

"To double-check," as a verb, means to look at something again to make sure that it is correct, or to confirm something.