Mało needs genitive?
Yes, you're right.
Can't I use "pory"?
No. "pora" is some specific time (like "wieczór" is "pora dnia" - evening is a time of the day), it's not used to measure time.
can you say, Mam bardzo trochę czasu.
Not really, or as a strange language joke. It's like saying "I have very some time".
"mało" is adverb, right? Why is an adverb used to serve as a direct object?
I'm far from being an expert on syntax, but I believe that here "mało" is an indefinite numeral. But yes, it can also be an adverb.
Would it be correct to say "Mam za mało czasu?"
It's a correct sentence that means something different, i.e. "I have too little time".
"very little time" may still be enough time.