So an M-113 Salt Vampire actually shows up on the ship 100+ years after they’ve gone extinct and there was no way for it to get aboard?
Yes. It turns out they were wrong in The Man Trap about their extinction. This has already been hinted at by their inclusion in a Starfleet Science video (Ephraim & Dot), and, noncanonically, various adventures in Star Trek Online. The M-113 creature in TOS was smart enough to get on board, so it stands to reason that others might be even more clever in their travels through the galaxy.
A drunken officer cuts off the limb of another with a Klingon sword and there’s no consequence?
Sort of. Just sliced his leg pretty badly. Probably an easy fix in the 24th century, but still a dangerous occurrence. There's a reason why Mariner remains an Ensign, and Boimler (in that very same episode) refuses to report on her actions.
Cmon guys, I don’t care how much enjoyment we get out if the show. None of it is meant to be taken seriously in the context of the larger Trek universe and continuity. It’s ridiculous. That’s what makes it fun.
McMahan says it counts, that it's all Star Trek, and that was his assurance to create the show. TOS, TNG, DS9, Discovery all had ridiculous, comedic episodes meant to be good fun. LD just has more on average than the others. If we suddenly start to discount LD episodes for jokes, then we have to throw out the Tribble episodes, I, Mudd, Magic to Make the Sanest Man Go Mad, most Ferengi episodes, half of the DS9 mirror arc, some of the Reg Barclay stuff, and much more.