A rare view of Russia's floating nuclear power station

They don't. It's going to supply power for arctic exploration projects.

The Russians weren't the first to float a nuclear reactor. See also, nuclear submarine, nuclear aircraft carrier.

Each Nimitz class aircraft carrier has two A4W reactors on board capable of generating over 200 MW of power with a service life of 20 years. That's enough to power around 60,000 homes. In the video, they mention supplying 45,000 people for 12 years, but no mention of MW output.

It's a good idea, but a couple of the problems you mention aren't exactly solved by floating the reactor in the ocean.

2. In the event of a failure, yes, only the people on board would be injured in the blast, but that's the case with most land-based reactors as well. Reactors don't blow up like nuclear bombs because the reaction isn't as focused. The fizzle out with a big glow and a huge emission of radiation. Bottom line is that it isn't all that much different from a land-based reactor, unless said reactor is downtown in an urban center.

3. I can't believe you're even suggesting that. Don't you think there is some reason we don't package up all our existing nuclear waste and dump it in the ocean?

The component of reactor failure that affects the most people is fallout. When a reactor fizzles out, it spews massive amounts of radiation in to the atmosphere. That radiation attaches to dust particles, which can travel hundreds, sometimes thousands, of miles. With a nuclear reactor floating around in the arctic, we have good reason to be concerned. The arctic is the shortest path from Russia to the US. The question is, what are the atmospheric patterns in the arctic? I don't know, but I'm definitely looking in to it.