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Some key insights from the article:
Basically, what they did was to look at how much batteries would be needed in a given area to provide constant power supply at least 97% of the time, and the calculate the costs of that solar+battery setup compared to coal and nuclear.
He is probably referring to the small amount of nuclear waste that is actually produced per watt of power, it is a lot more dangerous if you are in direct contact, but it is surprisingly easy to store safely, and remove all environmental impact. The biggest environmental issue with nuclear is the mining and enriching, both of which are realistically too small to factor in.
I found this article going into more depth nuclear waste .
No, none of that has much to do with CO2 output besides transportation.
Nuclear power needs a lot of concrete. Concrete releases a lot of CO2 during production. It does eventually reabsorb it as it cures over a decade or two. IIRC, it might even be CO2 net negative eventually.