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.
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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 satellites 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.
EDITING to add that there is now an answer elsewhere on this site re the bulge in the crust: How viscous is the Earth's mantle?
Changes in tides & ocean currents
If the planet were not rotating, the dominant period for tidal cycles would likely be related to a lunar month rather than to a day. There would also be no Coriolis effect, and these two factors 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 :-)
This is covered in an episode of the National Geographic TV series Aftermath called "When The Earth Stops Spinning". It's also covered by "If the Earth Stood Still: Modeling the absence of centrifugal force" by Witold Fraczek of Ersi, a GIS software company.
The Earth is not round, but bulges at the equator. The diameter at the equator is 43km more than pole-to-pole. Without the Earth's spin to force the oceans "uphill" at the equator, they would flow "downhill" towards the poles. Since the oceans are no where near as deep as 43km (the deepest part, Mariana Trench, is only 11km), the equator would be completely dry and the poles likely flooded. Aftermath depicts an Earth with two polar oceans and a strip of dry land at the equator. Most of North America, Europe, Northern Asia, Argentina and Antarctica are under water. Here's a timelapse video of that process.
Worse, the atmosphere acts like a fluid, too. It was also being held in place "uphill" at the poles. It also begins to thin out around the equator, thinner than on Mt Everest. Most of the world is either under water, or too high up for us to breathe. Humanity is left with a thin band of habitable land at the edge of the polar oceans, with some of the lowlands of the newly dry Pacific Basin also being habitable.
Doesn't matter, because with no spin a day on Earth is six months long. Baked during the "day" and frozen during the "night", it's not a pleasant place for people.