Some other world encounters say something along the lines of "Pass a Luck (-1) check. If you do, you find the gate back, return to Arkham."

Does this mean you can know close the gate, or spend 5 clues to seal it? Or does it simply kick you to a square that has a portal to the realm you are in, and you have to go back through. Because if that's the case, it doesn't seem very "lucky" to me.

The former, meaning it is lucky. This is answered officially in the faq. The exception is if the card specifically states otherwise. I don't think there are any other exceptions, but i could be wrong. If you want the faq, look at the arkham horror home page. Dude... Duuuude. Duuuhuuhuuuuuude. Dude. Yep. Dude.

As Minq mentioned, the former. Refer to FAQ on that.

However, take note you cannot close/seal on the same turn you return to Arkham. This is due to the OW Encounter phase being after the Arkham Encounter phase (the only phase where gates can be closed, sans specific encounters). What's worse, your movement immunity against monsters will wear off the turn you try closing/sealing, so sometimes it's actually better not to have that encounter, especially if you're at the second area of your OW.

Minq said:

There's one that says you close the gate but are Lost in Time and Space. That one doesn't let you seal the gate.

So if you get kicked back into Arkham by a OE card you have to fight the monsters there before you try to close it? Why are do you lose your movement immunity that turn (and by movement immunity I assume you mean the monster cannot attack you the turn you come back?)

The immunity lasts until the end of the turn during which you return. If you get a return to Arkham OW Encounter, you return during Phase 4. Basically, your immunity lasts rest of Phase 4 and Phase 5 (Mythos), after that, it's a new turn, immunity gone. Thus, during the new turn, Movement Phase (Phase 2), you have to deal with the monsters as normal, before getting a chance to close/seal during Phase 3.