Also, the average humidity is quite stable on earth, and any excess water that evaporates into the atmosphere quickly rains out again. Furthermore, the amount of water in the atmosphere is very small in comparison to the amount of water in the oceans. (See e.g. here: http://water.usgs.gov/edu/watercycleatmosphere.html). So it is very difficult to imagine that an increased humidity would have a significant effect on the sea level.
Look at this graph. By having the ice melt you'll have more salty water. Even if it is all fresh, the density maximum is close to 5 degrees Celsius. Ocean water is warmer in most of the oceans.
Water vapor is a greenhouse gas - more water vapor will mean even higher temperatures, which would lead to more melting.
The Arctic ice is important for albedo reasons, it reflects a lot of sunlight that would otherwise go into warming the water, but as you correctly point out, the melting of that particular ice won't contribute to sea level rise directly.
https://youtu.be/lPgZfhnCAdI?t=459
While looking for the clip I found this gem: https://www.youtube.com/watch?v=cjuGCJJUGsg