Weather in the Mediterranean

I can't speak to what the water would be like and whether that would impact your cruise as I am not a cruise-person. However, we were in Paris (pretty far from your area of interest) and Barcelona (more similar weather) in early April of this year and they were both just fine. Paris was mid-60s and a little spotty rain. Barcelona was low 70s and nice. Perhaps you can extrapolate out a bit from that info for your question about the Mediterranean coastline.

The weather that time of year should be fine. But their are never any guarantees with the weather.

Depends on your priorities. If you want to be sunbathing and frolicking in an on-deck swimming pool, weather will likely be too chilly. If you want to avoid peak-season costs and crowds, and you don't like hot weather, then early April would probably be ideal.
You can see historical weather patterns for various locations on the Weather Underground website.

I would expect it too be cool, especially at sea, and probably a little rainy. If it was me, I would opt for May as a better time. April would still be dealing with shorter days.

I wouldn't worry too much about longer days. You'll probably have to be back on board the ship before sunset anyway. Easter Sunday is April 16. Holy Week may impact something. Depends on the exact dates and port stops.

Weather is pretty good in April but not necessarily hot (18°c), as March is quite rainy.

In May you maximize your chances of having good and hot (25°c) weather.