Maps are here:

https://topex.ucsd.edu/grav_outreach/

Yes, water does follow the gravity. I'm not a climate person, and it's a complicated system, but as I understand it, new melt would be distributed evenly by mass around the globe, which means a larger volume of water at the equator due to higher temperatures there. My hunch in that changes in wind patterns and currents will be the more important effect though.