1. If I'm in an OW and I fail an Evade/Combat check against a Nightgaunt and there is no gate in Arkham linked to my OW, do I get LiTaS?
Yes, I would assume so.
2. If I fail an Evade/Combat check against a Nightgaunt in a location where I have an open gate with an explored marker, what happens? Do I go through that gate? Do I pick the second closest to me? Or do I simply end the combat and remain there?
You would be thrown back through the gate (thus losing the explored marker).