Is there another way to use the teleport gates than to pass the levels and several loading screens? Maybe using the map?! This would be very convenient, but if so, I did not yet find out how to do this...

When you're on the map screen leaving a place with a teleporter, you can hit the teleport option after selecting a destination with a teleporter. But then you still need to walk from the teleporter to where ever you're going within that location. It's definitely not a smooth operation. And taking the teleporter to a place close to your destination saves very little time no matter what it looks like on map overview page.

Unfortunately not, you just have to walk. As Mimo noted, it doesn't save much time, though it may spare you random encounters, food use in transit, and so on.