It'll depend on the available tools, materials, hardware, and how much time, effort, and money you want to put into it.
I build my bed frame from 2x8s, which happened to be the exact height for what I wanted. Basically all I had to do was cut them to length and cut out some notches on the ends so I wouldn't stubb my toes. Mine are in a rough egg-case pattern and open at the sides to allow storage boxes to slide underneath, but you could just as easily make a rectangle. I used angle brackets and screws to hold everything together so it can be taken apart later. Admittedly it is very rough but I just throw a bed skirt over the boxspring and everything's hidden.
I'd probably handle your height adjustability by using shims. If you wanted to get real fancy you could use slotted holes or bolts in the edge of the boards, but that just adds complexity.
Carpet underneath to prevent scratching the floor should work fine, but I would also consider felt furniture pads (like you put on chair legs... look in the hardware section)