No it's not. You can tell you're rotating using internal sensors alone, unlike linear velocity. Just because the magnitude of the earth's rotation is small compared to our biological sense of rotation doesn't mean rotation is relative the same way linear motion is. You're drawing a false connection between the fact that you can't tell if you're accelerating under gravity or not to rotation, which is fundamentally different. Also you don't rotate with units of linear velocity, but with units of angular velocity. The earth rotates at about 0.0042 deg/s, which is very slow. You rotate many orders of magnitude faster than that rolling over in bed or turning your head.

The universe having a nonzero total angular momentum does indeed imply an axis of rotation, but our theories don't explicitly rule that out. Given the size of the universe, the rate of rotation would be inconceivably small compared to the earth, and extremely difficult to measure. Most of my problem with your statements on this point is you're assuming current hypotheses to be confirmed fact set in stone, which is not true.