Firstwinter, renter's insurance typically covers a bike under personal property, but the catch is exactly what you ran into—the deductible often wipes out the value of a used commuter bike unless it's a high-end model. Some policies have special limits for theft away from home, and some exclude bikes used for work. If your bike is worth more than a couple hundred, it's worth a separate rider or a personal articles policy. For a used hybrid, probably not worth a claim.
On the gravity stand: yes, it can be very stable, but setup matters. I've used a heavy steel leaning rack for years, and I put a non-slip rug pad under the feet. That keeps it from sliding if you bump it. The bike's weight actually presses the rack into the floor, so it's not like a drying rack that topples. If you have a skittish cat, just make sure the bottom wheel is locked into the stand's cradles and maybe position it in a corner.
Big_city_beth, if you go the floor stand route, get one with a rub protector for the wall, or stick a cheap adhesive felt pad where the handlebar would touch. And if carrying up four flights is killing you, consider a lightweight folding bike eventually—not now, just when you're sick of it. Otherwise, a shoulder strap and wiping the tires will save your walls and your super's nerves.