No ... days get warmer or colder, drier or wetter, depending on where on earth you are.

No. The day is measured by 24 hours, i.e. 7.94243385x10^14  periods of the radiation corresponding to the transition between the 2 hyperfine levels of the ground state of cesium 133. Since this period of time will not be affected by the climate of the earth, the day will not get longer.

The answer is yes. The predicted maximum sea level rise this century is around 1 metre with about half due to thermal expansion and half due to melting of land based ice. This implies a 1 metre increase in the Earth's radius which amounts to about 1 part in 6 million.

Moment of inertia is proportional to the square of radius so, to conserve angular momentum, we could expect the Earth's rotation to slow down by about 1 part in 3 million. Taking into account the Earth's mean density of 5.5, this effect would be reduced to about 1 part in 16.5 million. This amounts to about 5 milliseconds per day or about 2 seconds per year.

The effect seems to me to be somewhat larger than expected. I'd be grateful if someone would point out where I've gone wrong!