Does entering a fight prevent use of cards with effects during movement phase afterwards?
No, it is not against the rules, and the sequence of events you followed is acceptable within a single Movement Phase.

Being drawn into combat of any kind only ends your movement - it does not end your Movement Phase, and some actions are still valid after combat.

While there is no specific case regarding the Key of Tawil  At'Umr, it can be inferred from an entry in the Arkham Horror FAQ regarding the Find Gate spell in an identical situation (on page 7 under Spells):


  Q: Can an investigator who entered the Other World
  during the Movement Phase because of the Nightgaunt’s
  combat ability immediately cast Find Gate and return to
  Arkham, experiencing no Other World encounters?
  
  A: Yes.


Neither Find Gate nor Key of Tawil At'Umr have any other pertinent trigger details except that they must be used during the Movement Phase, so it is logical that if one is valid after a Nightgaunt effect, the other is also.

Also on page 9 of the FAQ under Gates and Other Worlds it clarifies reasons why you would (or would not) gain an explored marker:


  Q: Do I receive an “explored” marker if I enter the first
  area of an Other World, cast “Find Gate,” and return to
  Arkham?
  
  A: Yes. In general, whenever you “return to Arkham,” you
  reappear at a gate that leads to the Other World you were
  in and gain an “explored” marker.


Since the Key of Tawil At'Umr specifically uses the phrase "return to Arkham" you do gain an explored marker when it is used in an Other World.