Is repetition the origin of countability?

Question

The original question was "Do all non-regular languages have an uncountable number of strings?".

How can someone prove that..? I am squeezing my head but I can't figure it out.

And the other side of the coin: is a language always regular, if it has a countably infinite number of strings?

A bit more generally: is repetition (DFAs) the origin of countability (and/or vice versa?) and if so, why?

Thank you in advance for your help.