Did some rigorous testing with this. Been a while since a Cave Story mod actually played really nice.
Found a lot of bugs though... Here's a list that will hopefully help you fix them.
General Bugs:
At several times, there will be an incorrect facepic, but then get's replaced with the correct one. (It usually happens within a second)
After obtaining the Slingshot, it doesn't give you the message saying that you refilled your ammo. (It does refill, just doesn't say that it does)
Curly faces the wrong way when operating the scooter.
Jukebox Area
Last 2 signs didn't play music. (Unless I didn't wait long enough)
You can jump off if you really want to...
Balcony/Prefab House Area
Checking the right-most bookcase will give you booster's note about the booster item and it's development. It also uses the "Provided if I live long enough." line.
When you enter the prefab house, you spawn up in the air as the new map doesn't have ground where you normally come in from the balcony.
When I jumped into the demon crown cutscene, I believe I ran into the pillar and hit some collision. (This also happened with the shack where you fight Igor. It's nothing major, but it feels weird to hit something when jumping)
Start Point/First Cave
In the Start point if you reload, a Curly Brace will be standing where you entered the map.
There was a little weirdness entering Mimiga Village... (Seemed off more like it)
If you return here with the door from Mimiga Village, you appear at the top before you entered the door.
Mimiga Village
Checking the top-most building will make some unreadable text (Text box never shows up) appear, and then you somehow unlock it with a key. (I didn't get the key) After this, it sends you into the Shack.
When leaving said shack, you appear behind King.
After talking to Sandaime, I found out you can ride the scooter without telling King at all. (If you go via bugged building)
When talking to Sandaime's mother, there's an akward spacing.
When you try to talk to Kanpachi, it doesn't work. (He says nothing and it's like a <END command when you interact with him)
Apparently, Jack's parents are no where to be found. (Same with King's parents)
Grasstown
It is possible to reach this area weaponless.
In Grasstown, you should make it so the critter doesn't see you. When enemies attack as soon as you go through a door transition, it might scare them.
After beating the enraged Mimiga in Grasstown, it gives the "Defeated Igor" message. I don't think that's supposed to be Igor.
When checking the scooter to go back to the shack, it should be the freeze command. During the choice to go back or not, it makes time run again. So if you didn't kill the critter, it'll hop around you, but do no damage.
In Grasstown, if you aren't careful, it's possible to use all of your ammo. There is the refill in Santa's house, but still... (Nearly ran out against Igor)
You can check the computer first and it has a blank text box when you had the choice.
Post-Demo Bugs
If you go back to the First Cave, you can get the flamethrower from tap. (Since this is after the demo ends, this isn't really a problem)
Bubbler has a messed up sprite in the inventory when you get it from the fireplace, but it works normally.
HUGE BUG: It is possible to get to the Graveyard and when you leave it, you get stuck in a wall; Forcing you to reset.
If you talk to Ma Pignon, you can't fight him.
Getting Arthur's Key before you get the shack key will appear as Arthur's Key in the inventory. (I believe)
Getting Arthur's Key after you get the shack key will not change a thing. (Except for getting stuck)
I took a couple pictures of some bugs if you need some reference. (Don't feel like embedding 7 separate pictures, so I'm gonna link the whole album here)
After all that bad news though, I do have some good things to say about this:
The jump eccentric parts actually bring some challenge and feel rather nice.
The killing enemies and getting health 100% of the time thing is actually really cool. (Too bad we don't see level 3 flamethrower and level 3 slingshot...)
2 Unique weapons. (Slingshot and hidden Flamethrower)
After you finish the demo, you have free exploration over what is included in the demo. (You can explore the first cave with the booster for example instead of kicking you out or leaving a black screen)