First question: Why did Jesus command the winds and water to be still there and then?
Second question: Why did the unclean spirits ask Jesus to be sent into the swine?
If they're fallen angels, don't they have heavenly bodies and isn't it weird to ask Jesus to be sent into pigs?
And if they're not fallen angels, what are they then?

The theory, which i read on some website on the web, goes as follows.
The whole storm suddenly dropping on the boat was actually the work of fallen angels and was intended to scare off Jesus and His disciples and possibly meant to kill Jesus prematurely.
When Jesus commanded the winds and waves to be still, He commanded these fallen angels to stop their attack through wind and waves.
They obviously didn't want Jesus to get to the possessed man.

The unclean spirits, also called devils, are in fact the free roaming spirits of dead nephilim and as they have no body, they roam the earth to find and possess a new one.
They are bound for hell and can boost the strength and power of possessed humans, as we read in those verses about the possessed man breaking chains and shackles. [That's why i suppose some martial arts gurus have superhuman strengths and abilities, because they are possessed and therefore gain those abilities]