To American ears, "toilet", when used in mixed company (a group of men and women)
 is very specific, indelicate, blunt, unsophisticated and crude.  I would not ask a host if I could urinate or defecate in the room of his house dedicated to bodily functions, but that is what is being asked when I say, "May I use your toilet?"  I'm not asking to take a shower or shave.  A man asking a male friend among men, fine.  A man asking anyone while in mixed company , no.  True, most activities in the bathroom are related to urination or defecation, but to my ears the question shows no class at all.

Women often say "I need to powder my nose."  Nobody really believes that they disappear (usually with another woman) to apply make-up to their noses, but...who knows?  It sounds polite, cultured.  Men sometimes say they want to wash their hands.  In fact, in a non-residential setting, they may say "lavatory" instead of bathroom.

As for as "Please no swimming", I don't understand the joke. 

I have, however, seen signs that have read "Welcome to Our Ool.  Notice that there is no P in it.  Let's keep it that way."  The joke is that pool is being spelled ool.  The name of the letter P is the same as an informal way to say urine:  pee.

Also, I have seen signs reading "We don't swim in your toilet, don't pee in our pool."