When we throw or drop an object while moving, do we impart it our acceleration or velocity or both?
Once you release the object the only force (neglecting air friction) on the body is gravity (force downward). Its horizontal motion will be its horizontal velocity at the release point. There is no horizontal acceleration after release.
Say i throw a ball at 5 m/s(relative to me) while at say, 30° to my direction of motion, then will the ball's speed in the ground frame be the vector addition of my speed and it's speed or will it be just 5 m/s?
Assuming your motion is horizontal, and that you are not moving near the speed of light, yes it will be the vector addition.
Consider another case in which i drop an object outside from my car window, will it continue accelerating with my cars acceleration of will it just possess the instantaneous velocity of my car at that instant?
It will have no horizontal acceleration once released since there is no horizontal force acting on it (again, neglecting air friction). So when you release it it will have your horizontal velocity at the time of release and initially no vertical velocity unless thrown downward. It will have a downward acceleration of $g$.
In any case, what causes it to retain any information about it's earlier motion? Why doesn't it just drop?
It's called Newton's first law, a body in motion will continue in motion and a body at rest will remain at rest unless acted upon by an external force.
Hope this helps.