I'm reasonably sure satellite imaging happens pretty far below geostationary, though. It'd be interesting if we could someday park satellites in geostationary and just stare at a given spot continuously, but lens technology just ain't there yet, last I checked.
Yes, they will. The only place you can put a geostationary satellite is on the equator. In theory, they stay in position above the selected point on the equator. In practice, they aren't perfectly stationary, and they will move around the designated point, which will, of necessity, cross the equator.