Fighting and rain are both nouns, so basically you use whichever noun you have had two weeks of, no matter what it ends in.
With the second example it's therefore wrong to say "after two weeks of fight" because you are trying to say that there was continuous, generalised fighting over a sustained period.
"The fight went on for two weeks" means one particular fight lasted for two weeks. You would then have to rephrase this as "after two weeks of the fight".
A noun with the -ing ending formed from a verb is called a gerund, and there are many examples — see ending in this very sentence. Some of them are so common that we don't notice them, like ending, but others stand out as more unusual.