Let's break this down:
"So you're telling me that an ultra sharp sword isn't capable of cutting through rotting flesh (or anything else apparently) unless the person holding it has been trained? It's a slashing weapon... Yeah, and what? Last time I looked they have pointy ends. Next you'll be telling me I need training to stab someone with a screw driver."
No, I'm telling you (for about the 10th time) that the katana is a weapon that cannot be utilised the way it should be - as a slashing and cutting weapon - since you chose to stand in a doorway with it! Then I'm telling you that if you want a stabbing weapon you'd actually be better off with the screwdriver since it is light, straight and only about 6 inches long, and is therefore easier to put where you want it to go, as well as being easy to retract quickly. A katana on the other hand is far heavier, curved and about two foot long, meaning that you are highly unlikely to put it where you want it to go with any degree of accuracy or force (when you bear in mind that you target here is a brain which will present itself as a target of about four inches wide and three inches high from the front) and also meaning that it is far likely that you're not going to get it back again the first time you screw up and embed it somewhere in the zombie's jaw.
"Care to let me attack you with a sword whilst you amble towards me at 2mph? I'm fairly sure you'll realise it's not going to go well for you, even if I was a 2 year old, short sighted toddler."
At 2mph a zombie would be over the 6 inch threshold of your doorway in under a second. This means that you have basically one chance to stab it (since we've established that only an idiot would try slashing through a doorframe) through the skull and into the brain forcefully enough to stop it completely. As also already established, that's probably just not going to happen. Moreover, even if you weren't so badly positioned in a dooorway I still doubt that you'd be anywhere near being effective at stopping this zombie because, as has been bought out here by several people who know more than you on the topic, doing it with any degree of effectivess requires skill. You would have to litterally cut the thing's head in half or off to stop it. This requires cutting through both flesh and bone, and your katana can be as 'ultra sharp' as you like but unless you land it exactly on the point of the sharp edge with the weight of the blade following correctly behind it (something which does require skill and training) you're most likely just going to find that you just keep smacking chunks out of the zombie's arms (at least until the blade dulls, as it will after the first few times it hits any bone) and that the zombie just keeps coming. The best you could hope to achieve would be just to smash its head in through repeated heavy blows with your piece of metal, and, again, there are far better bits of metal for that job than a katana. None of which work all that well in a doorway.
"I'm fully aware that to use it properly takes skill and training, but hacking shit up sloppily is insanely freakin' easy and zombies are hardly elite warriors from Feudal Japan."
Except (for the 11th time), you won't be hacking anything up in a doorway.
"Unless I missed that movie?"
I've already told you that they've never made a movie where anyone hacks anything up with a katana in a doorway; that's how stupid an idea it is. Even Jason Statham turned it down.
"Also, unless you're autistic and have to take things literally by definition of your condition, I think it's fairly obvious that I mean for them to be killed ahead of the door, so they fall into it on the floor and pile there."
So now you're not killing them in a doorway, you're just bascially stood outside a building? You're also still using your katana I assume, which is a weapon that is going to need about four foot of space minimum to wield effectively. So now you're four foot away from a doorway, the zombies attacking you can easily surround you and (here comes that magic again) the ones you kill are now apparently not just falling down four foot from the door (as the laws of physics suggest they would) but actually helping you out by taking themselves over to the doorway and lying down atop one another (moving a little bit to let you pass when you're ready to head inside I assume?). And if, at this point, you're thinking: "yeah but I'd be walking backward while I killed them so that would create a pile up as I moved through the doorway!" No (for the 12th time) as you walked backward you would once again find that you are in a doorway and so can longer wield the katana.
FINAL RECAP OF YOUR PLAN:
1, Stand four foot from a doorway.
2, Hack wildly at approaching lead zombie.
3, Have trouble getting its head because of its outstreched arms which aren't flying off like they do in the movies because you have no skill with your katana, which is also now basically blunt.
4, Hack even more wildy as you realise other zombies are now surrounding you.
5, Luckily smash the first zombie's head and then look confused as it falls sideways, nowhere near the doorway and leads to no sort of 'pile up.'
6, Retreat back toward doorway.
7, Swing at next zombie.
8, Wonder where sword has gone and realise it is now stuck in the doorframe.
9, Piss pants.
10, Run through doorway and manage to actually retrieve sword as you go.
11, Try to stab zombie.
12, Miss.
13, Try to stab zombie.
14, Miss.
15, Try to stab zombie and manage to drive the sword through its cheek where sword becomes lodged.
16, Shit pants.
17, Run away (provided you have entered somewhere with another exit).
18, Find new pants.