This works because anything which floats upon water displaces precisely as much water as its mass. Since ice is less dense than water, it floats, and some of it remains above the water. But because the ice displaces exactly as much water as its mass, when it melts you're just left with water at the same level.
Stay the same. Sea level rise will not occur because of the melting of sea ice. Rather, it occurs because of the melting of ice over land.
However, there is a secondary problem with respect to sea ice melt: sea ice tends to be very white, while the sea is very dark. This means that when there is less sea ice cover during summer months, the arctic sea tends to absorb dramatically more sunlight. This is why the Arctic sea is the one place on Earth that has warmed the most in recent decades, with some areas being as much as 10C warmer (on occasion) than they were 40 years ago. The antarctic has also seen a significant rise in temperature for similar reasons.
And this can have disastrous consequences for the ice locked on land in the adjoining areas, such as Greenland. The loss of ice in these areas has been accelerating in recent decades, with recent measurements showing much more ice loss than any models have predicted.