Why is it "det regner" instead of "Den regner"?
det is used when referring to a neutral-gender (t-words) word, while den is used for commen-gender (n-words) words. But if you are not referring to anything speciffic, you just use det
Det kan ikke regne hele tiden.
What does the alternate meaning for 'regner' (Is calculating) mean? What is the connection with rain and calculations?
That is not a semantic connection but a homonym. Bad luck. The words happen to be very similar in my language too, apparently here they are even more similar.
"DET REIGNER MÆND, HALLELUJAH!" (jeg undskylder)
Vær forsigtig, Simone! :O