"happen to be" and "happen to have" - expressions of probability?

"Happen" does come from the Middle English word "hap," meaning luck or chance, and it does retain that connotation in situations where it means occurs by chance:

I just happened to meet my cousin at the concert.

We didn't plan to go to the same concert. It was just a chance meeting.

But "happen" also simply means occurs:

The scandal is called "Watergate" because that's the name of the building where the break-in happened.

No chance involved: the break-in was deliberate.

Sometimes the word is employed for things that themselves didn't come about randomly but are relevant to the conversation only by chance:

A: My program isn't working. I wish someone could help.
B: Let me take a look. I happen to have a degree in computer science.

Here B is saying that he has a degree in computer science, and this is something that doesn't come about by chance but by planned study. What is a chance happening is that A needs help and has run into someone who can help.

In all these cases (and in your examples as well), these are statements of fact and not probability.

No, not at all. In both cases, something "happened". It is a fact, not just a possibility.

To say "happened to be" or "happened to have" implies that it could have happened by chance (not likely for a degree in computer science—that would necessarily be intentional).

It does not include an iota of chance that the thing might NOT have happened.