Takahide

Does Takahide choose which gold card to start with?

Look at the text of each of those cards.

For takahide, are the gold cards he switches to visible to other players when he switches?

It depends. If it was facedown, then no.

With Takahide's alt red (the one that blocks spaces), can Pyro end a move adjacent to Takahide? Also, can Arien's gold move Takahide after Takahide's red?

Follow the text verbatim. Is it something done by an enemy hero? Those spaces are obstacles. See "Obstacles" in keywords section.