True and I'm an idiot for this one.....

I guess she thought that was a convenient place to hide the gun: she was staying in the studios and she knew or was afraid that the police, after the discovery of the body, could search all the persons in the building. So she had to conceal the gun: does this make sense?

I agree. When she put it there it was not originally visible.

Hi
Seeing robert culp as many different baddies is excellent,seeing his reaction evrytime Columbo catches his, his just great.

Kay Freestone have a very limited amount of time
to dispose of the gun, so the elevator was the
best choice.
Kay made the same mistake that Hayden Danziger
made...they carried out the murder in such a way
that only a limited number of people could be
possible suspects...in Kay's case, the people in
the building which was well guarded from the outside,
and in Danziger's case, the people on the ship.
It is true there were several hundred, but it still
is possible to eliminate most easily. Danziger should
have tracked her down in Las Vegas and done it there.
Ward Fowler was really stupid, because if he hadn't
opened his mouth and let Claire Daley recognize him,
Columbo would have been faced with a murder that
could have been committed by anybody.
One murderer who realized this problem was Dr Bart Keppel in "Double Exposure", security was not tight in the building, but there were too many questions, e.g. why shoot the victim at the water cooler, how would they know when he would come out, why did Keppel keep the tape recorder going, etc.
Well, no one is perfect. As Sgt Hoffman stated in
"Negative Reaction": "Everybody makes mistakes..that's why the prisons are all full up!"