If the earth stopped rotating on its axis, would this influence its revolution motion? In particular, could it collapse on the sun? I ask this because on the one hand I thought the two degrees of freedom (rotation around the inner axis and revolution around the sun) were separated, on the other phenomena such as tidal locking seem to show a clear influence of one on the other.
The coupling between rotation and revolution, that leads to tidal locking, is usually very weak so it takes a very long time. So in the short run, you could give the Earth any rotation you like, without affecting its orbit. However, in the long run, there would be orbital effects. Right now, the Earth is spinning more rapidly than its revolution rate, so angular momentum is being stolen very slowly from the Earth's spin, and put into its orbit. But it wouldn't matter much, there's not much angular momentum in the spin compared to the orbit.
Rotating around itself is one thing and rotating around another object is something else. It is perfectly fine for the Earth to not rotate around itself while keep rotating around Sun. It is the "revolution" around Sun that keeps Earth not to fall into Sun. The answer in short: The condition mentioned would not influence the gravitational binding between Earth and Sun. Thanks,