There is a gate card that has the player roll a check to either leave the other world, and return to Arkham on a success, or on a failure is Lost in Time and Space. Either way, they close the gate related to the other world.

When this happens, could the investigator spend 5 clue tokens to seal the gate? Or is the ONLY time you can seal a gate, is if you make a success check during Arkham Encounter phase to close a gate?

Also, is there a quest for Tycho and Gabe?

As posted in the FAQ thread, you may not seal the gate with this particular encounter.

No, Tycho and Gabe do not have personal stories. But feel free to draft a couple using Chris Jennings's program, Strange Eons.