Would someone with extreme cat allergies still experience allergic symptoms if the cat is in a separate apartment on a different floor?

Unlikely: Cat allergies are due to dander sticking to cat's hair, so even in the absence of a cat, you can have symptoms if the cat was staying in the same place and shed its hairs for some time before it is removed. Cat dander doesn't travel by air as pollens do. You need to avoid cats or places were cats dwell as much as you can, see an allergist/immunologist for desensitization if possible, good luck