I think the earth would also stop rotating, so we'd be stuck in either an eternity of day or an eternity of night, even an eternity of a certain season. But then we would still need a sun for that to happen in the first place.
I think it would speed up slightly in its rotation. When the atmosphere dissipated (which it would do faster than anything else, being the most tenuous and least dense part of the Earth), the rest of the earth would be left with a lower moment of inertia, and would therefore spin faster.
Of course that would just barely happen, because at the same time, the earth would be crumbling and flinging off it's outer layers into space, while the inner layers would be expanding outward to release the enormous pressure. It would explode in a way, just more slowly than the sun would (since the internal pressures would not be nearly as great as the sun's, lacking the nuclear reactions and extreme heat).
Oh, also we can redefine mass and time in terms of light.
Mass: How matter interacts with light: absorbs it, reflects it, or deflects it.
There is little correlation between mass and interactions with light.
Don't we define the universe of how much mass we know exists, and how it interacts with itself?
No. We need to consider energy and the fundamental forces as well. Mass itself is not that well understood; with matter broken down into to so many different subatomic particles, and having anti-versions, and likely being a sort of illusion created by strings vibrating in 10 or more dimensions, we don't have a very clear idea of where mass even comes from. It might be possible to have matter with no mass, or mass with no matter. (Stay tuned for the Higgs Boson and other intriguing developments to come from the Hadron Accelerator.)
Also, it's pretty well established that we don't know the exact size or content of the universe. When we say "universe," we don't just mean the "visible universe," which is all we can observe and measure.
But again we define mass as the weight of matter. Which then needs gravity.
No, mass is better defined as a measure of the inertia of matter. Weight changes depending on which planet you're on, but mass does not.