I don't know how to tell if there is enough room.
You may have to measure. If your current card is cramped for space, and the card you're getting is bigger, you may have difficulties.
Or enough cooling. Do I need to go to a card without a fan to be on the safe side?
Just the opposite. If you get a card without a fan, you have to be extra sure your case is well ventilated. A card that is only cooled by its heat sink is more dependent on the ambient temperature inside the computer case for cooling.