Mountain bike is fine, eventually you could put hybrid tires on it but try it for a while first.

Your bike is probably safer in your cubicle than it would be in a bike rack anyway. Certainly less exposed to the elements.

Shower: Everyone has their own routine, but the one trick I discovered that makes a big difference is wait about 10 minutes before changing. If I change right away, I continue to sweat a while. Some days, if I don't have an important meeting, I don't change, since I just wear regular jeans or shorts & a tshirt, not a biking outfit.