Does this sign mean both 'Russian' and 'Russia'?
Yes, in ASL, the same sign is used for both the country 'Russia' and the adjective 'Russian' (such as Russian people or Russian language). The context of your sentence will make the specific meaning clear to the person you are signing with.
Do I need to tap my hips a specific number of times?
Typically, a deliberate double tap is used to clearly indicate the sign. Tapping just once might look like you are simply resting your hands on your hips, while tapping too many times can look exaggerated. Two crisp taps are standard.
What handshape should I use for this sign?
Usually, the hands are in a relaxed '5' shape or slightly curved, with the palms and fingers resting against the hips. The exact handshape is less strict than the location and movement; the key is the placement on the waist and the rhythmic tapping motion.