Does bruxism cause your gums to itch?

Not that I know of. I have not heard of Bruxism causing gums to itch.
Itching gums. Good question. Some people describe (particuarly if teeth are damaged to root exposure) that there is a numbness of tingling of gums on affected side (this can be one or both). It is nto the bruxism iself but rather the damage to teeth and gum structures.
Probably not. Probably not. If the sensation concerns you, see your dentist for evaluation.
I wouldn't think so. I've never heard of that being an issue.
It could. When a person bruxes, they cause forces on the teeth that are transmitted to a ligament that surrounds the root of the tooth. This ligament's function is to act a s a shock absorber. There is build-in space for the tooth to move slightly in and side to side. It is this movement that causes varied sensations on the tooth. The gums could feel as if they need to be scratched.
Of course. Chronic night grinding or clenching of the teeth can often lead to such sensation. See your dentist for night guard fabrication.