Is there a way to make a quadcopter maintain steady hovering (no lateral movement, constant altitude) while tilted left or right? If so, how can i accomplish this?
In a normal quadcopter configuration, if you tilt you will be pointing some of the lift force to the side. The only way to hover at a tilt, then, would be to somehow apply a counter force. A standard quadcopter can't do that itself, so it would need be an external force (tether, wind, etc).
Your other options are to use a nonstandard configuration such as actuated motors that let you tilt them individually or adding extra lateral motors.
No. Once you're quad is tilted you are projecting the thrust of the motors at an angle which is not parallel to the z axis causing your quadcopter to move linearly along the x-y plane.
If you must hover while tilted, you can use stepper motors to keep the thrust projecting parallel to the z axis.
That would be an unusual quad though.