Will combat damage trigger fight abilities?
No, combat damage does not count as fighting, and will not trigger an ability that says "whenever a creature you control fights."

"Fight" is defined in the official rules. The text of it doesn't matter; all that matters is that it's a defined thing, and that it doesn't tie itself to combat in any way, so only something that actually says "fight" is a "fight".

Similarly, the entire combat procedure is spelled out in the rules. Nowhere in it does it mention "fight", so it definitely does not have any interaction with fight. (In fact, the only section of the rules that mentions fight is the section defining it, plus the summary of that definition in the glossary.)