How do they figure it is "desta" not "deste or este/esta?"
Because camisa is feminine. Desta camisa/ deste vestido
Should pois work here in place of porque?
Yes but is less common in spoken discourse.
This exact sentence came up 5 times so far in this lesson...
He really likes his blue shirt.
Why can't I write "esta" instead of desta?
You should use "de" after gostar. De + esta = desta.