Mon Mome was the racehorse who won the 2009 Grand National, which took place on the 4th of April... so 4/4/9
Unrestricted depends on the game, usually it's completing something without any restrictions on glitches, or a certain in-game item, or other exploits. Make sure to check the game's rules and category rules.
NMS, as far as I've ever seen, just means 'No Major Skips' and means one or many major glitches, exploits, or indeed skips, have been banned from that category but might be allowed elsewhere. Again, check the category and game rules for exactly what the category is forbidding.
If you do want to know exactly what's different about these categories for a specific game you can always try asking in the community, or a moderator directly failing that.
Welcome - very nice to meet you, good luck with your runs :)
Lots of games have extra categories involving getting all collectibles, or 100% categories involving that kind of thing.
Assassin's Creed's series is a terrifying example of that https://www.speedrun.com/series/assassins_creed
D.W.W.R - 'Do What We Rehearsed'
-xem92 (or possibly somewhere else in the Ratchet community) this is just where I've heard it, and honestly for complicated speedruns it's great and helps you stay level headed and avoid panicky mistakes.