Well, I certainly hope they've at least stopped issuing waypoints with identical names... although it wouldn't surprise me if geographically-distant is the best we can do as a species.
They appear to be sequences of 5 upper-case letters. Assuming the 26-character alphabet, that should allow for nearly 12 million unique waypoint IDs. The world is a big place but that seems like it should be enough. The more likely problem is that there is (or was) no internationally-recognized authority in charge of handing out waypoint IDs, so we have at least legacy duplicates if not potential new ones.
You have to reduce that to the (still massive) set of IDs that are somewhat pronounceable in languages that use the Latin script. You don't want to be the air traffic controller trying to work out how to say 'Lufthansa 451, fly direct QXKCD'. Nonetheless, I think the there is little cause for concern about changing existing IDs. There might be sentimental attachment, but it takes barely a few flights before the new IDs start sticking, and it's not like pilots never fly new routes.
No, waypoints aren't spelled out with the ICAO alphabet. They are mnemonics that are pronounced as a word and only spelled out if the person on the receiving end requests it because of bad radio reception, or unfamiliarity with the area/waypoint.
For example, Hungarian waypoints, at least the more important ones are normally named after cities, towns or other geographical locations near them, and use the locations name or abbreviated name, being careful that they can be pronounced reasonably easily for English speakers. Like: ERGOM (for the city Esztergom), ABONY (for the town Füzesabony), SOPRO (for Sopron), etc.
It is, but fixes are almost always spoken as words rather than letter-by-letter. For this reason, they are usually chosen to be somewhat pronounceable, and occasionally you even get jokes in the names. Likewise, radio beacons and airports are usually referred to by the name of their location; for instance "proceed direct Dover" rather than "proceed direct Delta Victor Romeo".
I think a lot of pilots and air traffic controllers would be irritated if they had to spend longer reading out clearances and instructions. In a world where vocal communication is still the primary method of air traffic control, there might be a measurable reduction in capacity in some busier regions.
Disney has a whole lot of special fixes in Orlando and Anaheim. The PIGLT arrival passes through HKUNA, MTATA, JAZMN, JAFAR, RFIKI, TTIGR. I'm fairly sure I've heard about some variants on MICKY, MINEE, GOOFY, PLUTO, etc.
