The answer to the ice cube in the cup of water can be easily looked up, and the answer is that the level would remain the same.

When the last ice age ended, the global sea level rose by about 120 metres. This was not due to the ice floating on top of the Atlantic, it was due to all the land ice that melted and flowed into the sea. There is only one significant land mass in the northern hemisphere today that may have a similar effect: Greenland. Otherwise juiceme is right: all the North Pole sea ice melting will have no effect on the seafloor below it, as it does not matter whether it sits there frozen or melted. The mass is the same.

The melting of ice age ice did have an effect of releasing some weight off the land below it and the land rebounded a bit. This may have indeed triggered a few local, small-scale quakes, the likes of we observe nowadays as for example linked to fracking. These are all small, shallow, local affairs, barely registered by anything other than instruments. Any tsunamis would also be local, caused by calved chunks of ice splashing into a fjord or some such. The quakes and tsunamis like we saw in Palu in 2018, Fukushima in 2011 and Indian Ocean in 2004 were deep, massive events caused by continental drift that does not care about such trifles as a bit of ice.