Translation:Are you able to swim in the swimming pool?
Pwll means pond or pool and (at least in my dialect) a mine can be referred to as a "pwll" typically with a substance coming after it i.e "Pwll glo" being a coal mine or more literally a "Pool of coal".
Now that is interesting. In the US, old quarries will sometimes become swimming holes, when they fill with water. I wonder if that is what happened in Wales as well.
Yeah, check out the Blue Lagoon in Abereiddi. It's an old quarry that's now filled by the sea and is used by adventure groups, swimmers and divers (though the signs advise against that last one as it's dangerous): http://www.pembrokeshirecoast.org.uk/Default.asp?pid=628&LangID=2
My original (snarky) thought was, "No, but I can swim in the fridge." On a more serious note, could this kind of sentence construction be used in a sense of whether people can generally swim in the swimming pool? For example, "There's a swimming pool at the hotel." "Oh...can you swim in the swimming pool?" In other words, something like a less formal idea of "can one swim..."?
Yes, it can. If you call the person you're talking to ti, then Wyt ti'n gallu nofio yn y pwll nofio? is the normal way of asking "Can you/one...". In some dialects chi is reserved for this "one", so Dych chi'n gallu nofio yn y pwll nofio?, but that's uncommon.
No, this is asking about ability (gallu/medru) to swim, not permission (cael) to swim.