This is correct. The earth is ductile at long time periods, and mountains maintain isostatic equilibrium as the tops are eroded away. The article quote is absurd. Source: my earth science PhD.
Guess I just went and looked at a Geoid map and I was lied to, EU is in big trouble!
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.