Can punching things and throwing yourself around but without hitting your head cause a concussion? and would going on a bumpy bike ride after that cause further damage?

No risk at all. Neither of these activities will cause consussion.

By definition, a concussion involves a closed head injury resulting in loss of consciousness, amnesia, or a combination of both. In absence of these, you have not experienced a concussion.

I am not sure exactly how someone throws themselves around but Traumatic Brain Injuries do not necessitate hit their head. There can just be a severe jerking motion such as whiplash from a motor vehicle accident. Have to be careful. Concussions are usually caused by a blow to the head. Violently shaking of the head and upper body also can cause concussions.