I have a Cube Analog Disc 2012 bike, and recently discovered that I have no screws inserted in the bottle cage mounts. Is it possible for rust to form in this area? Also, is it possible for the water to enter inside the tube and develop rust on the inside (since there is no hole at the bottom of the frame, I believe water will remain inside until it clears off)?
It is possible for water or sweat to enter the frame, and for corrosion to build up. Any rust would be most likely to happen on the steel bolt (which is missing) rather than on the aluminum frame, but oxidization comes in many forms.
While this is unlikely to cause you any significant issues, it's an easy fix. Any M4 threaded allen bolt, between 1.5 and 2.5cm will fit just fine.
Stop by the hardware store or your LBS, and you'll be sorted in just a few minutes.
Your bike has aluminium frame, it can not suffer from rust. There is such thing as aluminium corrosion, but it is a very slow process, you bike will be fine for years.
But, of course, you better cover the holes. Cheap solution is a piece of tape, but you can either buy a bottle cage or any screws that fit there.