I play ball hockey and roller hockey goalie. Only tried ice a few times so can’t help with the transition part.
As far as playing style I would say I play a pretty hybrid/butterfly style, I do challenge aggressively which leads to diving head first to cover back door passes. As far as standup I try to stay on my feet as long as possible before I butterfly, but I don’t have much trouble moving across the net once I’m down.
Our league plays 5v5, on sport court tiles the arena isn’t full sized. I have played in full size arenas with the ice out and small gyms very similar to what you describe, but I don’t really change my play style at all.
For gear I’m in full ice gear at this point, I did use player pants and jock until I started playing roller.
Hope this helps!