I see that I would have 40 badges, and should be able to select my favorite 5 from these. 

Yet when I click on the profile "badges", I can see about 20 and only about a dozen are unlocked. 

There is no pagination to see more badges. 

Are they somewhere else, or there is something with the UI here? 

Some of the badges are higher levels of existing ones, e.g. Survivor : 1 week badge, 1 month badge, 3 months badge, etc. So you have 40 badges, but some of them are stacked on top of others.

In order to select your favorite one(s), click on them and then on one of the 5 slots below your profile name / picture.