A lot of the listed criteria do not make it clear whether or not activity on private teams will give you a hat.
A few examples I am not sure if teams would qualify or not:
post a competing answer to an existing accepted answer, reaching a score of at least +3
earn 150 reputation points on three different sites (not including Stack Overflow) within 15 days.
Or do teams posts just generally not count in any way?
Are there any guidelines for which types of things you might get credit for in a team or not?