First thing would be to apply a lubricant to brake pivots. Given that the bike is cheap, its cables are presumably regular steel so putting some oil into cable housing might be also advisable.

As to the use of the right hand only for braking, the right lever commonly operates the rear brake, while the front brake is more effective. In the long-term you should aim at a changed arrangement on the bike so that the right hand begins to operate the front brake. It is something a mechanic should be able to accomplish.