Karthus is fine as he is. He's not S tier but he functions well. He has one of the best level 1s in the game and scales incredibly well, on par with some of the hardest scaling mages. If Karthus had mobility he could just kite in and out of fights using his Qs to deal damage constantly (20 mana for God's sake), which isn't really a good thing since you need to be able to get to him and kill him. If hitting his Q is a problem, it has almost no CD or mana cost, so you'll hit a few eventually. It's not like it's a Lux E or something with at least a > 1.5 s CD. Also his wall is meant as a soft deterrent, not a hard stopper. It functions for small catches and slowing targets. The slow is actually pretty strong considering it covers almost the entire width of mid lane at max level I believe (been a little while since I've played him). Karthus isn't bad atm. He just needs to hard scale, and games have to last long enough for him to do that.
Wukong I'm almost positive can't dodge it because it's nothing but an invisibility, and you can still be damaged during that. If he can, then Rengar ult and Talon ult should dodge it as well. As well as Shaco Q and Twitch Q. And LB passive. I'm fairly sure none of those work.
Shaco ceases to exist on the map for that half second during his ult activation so that's why he dodges it. Same principle with a Yi Q. Yi ceases to exist on the map for a half second.
Fizz's E makes him untargetable but he still takes damage from DoT. There's nothing wrong with that. Any DoT effects counter Fizz quite handily as he can't stop the damage. Fizz is also naturally frustrating in his own right as a champion, so that annoyance factor is just intrinsic in the champion.
It's not like assassins are the only ones who can counter Karthus ult. Kayle, Shaco, Zed, Fizz, Zhonya's, Morg (sort of), Lissandra, Kindred, Yi, Ekko, Vlad, Tryndamere (sort of), Sivir, Nocturne, Bard, Fiora, etc. can all deal with the ult.
Funny thing is, several of these can do the same thing with anything in the game. Any skill shot, targeted spell, etc. A notable one being Zed ult, which is much weaker than Karthus ult, I would argue. It's just a matter of a counter present in the kits/item builds of certain champions. If you couldn't get away from Karthus ult all it would be grossly overpowered. I mean, a spell that hits all enemies wherever they are is kind of silly if no counter exists outside of hitting Karthus himself.
There's nothing wrong with legitimate counters to such a spell.