Would sea level change at the equator if the Earth stopped spinning? I am assuming it is currently bulging around it due to centrifugal force.
Let's assume that the earth didn't suddenly stop spinning (because intertia and conservation of angular momentum would do all sorts of "interesting" things that are deserving of a What-If answer), and stipulate that the earth slowed down gradually, or possibly that it was never spinning in the first place (although I'm sure this would have all sorts of other effects that wouldn't have got us to where we are...)
Yes, sea levels would change, but not necessarily for the reasons that you think.
Part of the bulge in oceans is due to centrifugal force on the water, but much of it is not. There is an underlying bulge in the seabed as well as the ocean. A result of this (and other variations in the thickness and density of the crust) are that the earth's gravitational field is not even across the globe, and where there is a stronger area of gravitational field, more water is pulled towards it and a bulge results. It is this effect that allows for the bathymetry of oceans to be mapped by sattelites that sense the elevation of the sea's surface.
I am no geoscientist, but I imagine that this bulge in the crust at the equator is also to do with centrifugal force - but it would take a lot longer to go away, if indeed it did at all, than one caused just by water.
Changes in tides & ocean currents
If the planet were not rotating there would be no Coriolis effect, and this would result in major differences to tides and to ocean circulations. As such, it is likely that there would be substantial differences in both short- and long-term elevation changes that are due to currents.
I suspect that lack of rotation might have effects on the planet's core and its magnetic field, which might result in all sorts of other impacts... but we'll have to wait for a geoscientist in a speculative mood to talk about things like that :-)