Both are grammatically correct but idiomatically only the first one works. "es gibt" does not express well where living beings stay. It rather applies to dead matter and (more or less) permanent situations (there may be exceptions to this)."There's a cat in the room"
Is it possible to also say, "In dem Zimmer ist eine Katze" instead of "Es gibt eine Katze in dem Zimmer"?
This concept of "es ist" and "es gibt" is absolutely new to me.The difference is that "es ist" is much more precise than "es gibt", it doesn't have anything to do with whether it has to do with living beings. Both work fine most of the time, with one always sounding a wee bit unusual. In general, you state the exact position of something by "es ist", whereas by "es gibt" you just state the approximate place.
Hinter der Mauer ist ein Haus. (The wall is possibly pretty small and once you're on the other side of the wall, the house would instantly catch your eye.)
Hinter der Mauer gibt es ein Haus. (The wall is possibly huge and you cannot understand the exact position of the house from this sentence. You'd also have a hard time finding it if you were on the other side of the wall.)
As for your sentence, "es ist" would be appropriate, as "Zimmer" are usually pretty small and things could be find easily in them. If you used "Raum" of which the size could be small and large, it'd solely depend on its size. Classrooms (Klassenräume, "Klassenzimmer" is possible too) are rather small, whereas rooms in a museum are often rather large.