The square dimension symbol on the Mythos card indicates that the Hound of Tindalos' special movement ability is activated. I have only 2 Investigators inside Arkham locations but both are sitting on top of elder sign symbols signifying Gates that have been successfully sealed. Does the Hound stay where he is currently located?

Nope, the Hound moves. Elder Signs make a location stable, which means gates and monsters cannot appear. The Hound here resolves his normal movement, and moving and appearing are different keywords.