First: the 10,000 year mandatory storage requirement: not driven by science, fact, etc; driven by law. As I recall, there is nothing man-made (not piles of rocks) on Earth that old, but we wrote law to say we had to build something that would last that long. And prove that it would.

Second: If we really wanted to get rid of waste, we could - the oceanic subduction zones are perfect.
1. Shape all your high-level waste into chunks
2. drop it into a subduction zone
3.???
4. in a few years (profit!!) it goes deep into the Earth where it came from.
But, we (the US) have mandated that you must be able to check on your waste storage, ensure it is still there, happy, and no one else took it. So, lots of the good permanent solutions are out-of-bounds.

C. Yucca Mountain: the site was chosen by Congress, without actually completing a 'competitive' review. There were several sites under consideration; Congress picked Yucca, then told the DOE to perform sufficient studies to deem it safe. There actually is some evidence that surface water penetrates to the storage tunnel levels quickly (100s of years). (No, I don't have the link anymore; is from DOE site reports in the mid-90s.)

D. On-site storage: All the powerplants are currently storing high-level waste on-site; not in any 'secure' location. Why? Because the government declared that a high-level waste storage facility would be available in 1996....and yet we wait.

Lastly, somewhat off-my-own-topic: you can't usefully use a nuclear powered ship for electric power generation - a very small percentage of total power generated goes for electricity. It takes far more energy (in the form of steam, to turbines) to drive a hull through the water. Thus, if a hypothetical shipboard power plant was rated at 50MW thermal, it probably only produces 5MW in electrical power. You'd have to completely re-vamp the steam plant to dedicate the entire thermal output to electrical generation.

Disclaimer: Yes, I actually am a nuclear engineer. I've been running Naval power plants for 15 years, and spent a few years doing research on rad waste disposal. Just don't have any of my references handy (something about being at sea.)