It's a drop in a bucket. Global warming is due to ~5e18 Wh of excess heating. A nuke dumping 5e12 Wh of additional heat into water makes no difference at all.
No, they really don't in this case. IIRC a sustained gigawatt of output over a 24 hour period is roughly equivalent to the sunlight absorbed by a 5 square kilometer patch of ocean in the tropics over the same period of time. The oceans absorb an almost incomprehensible amount of solar energy on a daily basis.