Just finished the demo and I think this has a lot of potential!
I got stuck in the sunshine room too until I checked the comments, aside from that room I didn't have any trouble with the puzzles.
First things first, I like the old net vibe a lot and and the windows-based textboxes are cute! The character designs are also really nice, I love the way you do the eyes on your characters' overworld sprites! I also thought the swap in graphics from RTP to custom was used really thoughtfully!
My gripes stem more from the narrative being a little hard to follow with Arlen at first (though from the nature of RPG Maker horror games I know better than to trust her farther than I can throw her). If there was maybe more of a primer on their relationship during the beginning that could help set the stage (though it's possible something was in Melrose's room that I simply didn't check). However this can be a stylistic choice, it really depends on how clear you want it to be and if you don't want to give away too much off the bat (which I understand).
On the same note I have a small gripe with the tea party scene, not that it was a bad scene, but it would benefit from letting the game wait 3-5 seconds after the fade to black before resuming with the gameplay, just to allow the scene to set in for your player. It ended nicely but I was given control back so quickly I thought it was jarring! Cutscenes are your player's rewards in these types of puzzle grindy games, so take extra care there! (at least, that's my view. don't let my thoughts go against your vision!)
My remaining thoughts are more event coding polish. Like it would be nice if teleporting to a new map didn't place you right on the tile you'd need to step on to go back, since on the odd occasion I entered a room by mistake, it was cumbersome to take a step forward then back to leave, especially because of Melrose's slow walk speed (plus her dash isn't much better) and already getting a little sick of pressing enter to go into more rooms than I would have liked.
However personally I wouldn't increase her movespeed too much, or maybe make her dash her default speed instead if possible. Usually my issue with dash buttons in RPG Maker is that it's so fast I have no reason not to use it, I think you made a good call making the dash only a little better so it's used more thoughtfully (like during the spike puzzles). But in keeping it as is, I'd adjust your level design to be more accommodating to it or just make those teleport tiles less cumbersome lol. It really wore me down on a long playthrough.
Other notes are the ball of yarn reappearing after I took it if I left the cat map and came back before solving the puzzle (found this because I didn't realize I was supposed to talk to the cat that was hurting me, I mean why would I? A good visual indicator for it would be maybe to set it so it faces the player after you take the yarn so I can tell something's up. Though if it does and I missed it my b lol). I could tell the name of the game was check things twice and I believe the game even states this pretty early on but little cues and reminders can go a long way.
On damage, sometimes it felt a little defanged, especially if a puzzle damaged as a punishment while a rose was literally 3 tiles away (the quiz being the biggest offender). You could start the player with more HP but less healing spots to make it more meaningful, or just drop the punishment on spots where it just has no bite whatsoever. But that really depends on how challenging or thoughtful you want the HP to be.
I also didn't find the atmosphere to be... too creepy? But I do tend to write it off as myself having a fairly high resistance to it (especially having played Ib, Yume Nikki, Witch's House, etc in my teens). I'm not sure how eerie you want the game to be, some aspects come off more as campy to me ("having fun yet?" comes to mind). Where to go with it though depends on your vision.
Overall, I'm interested in where this goes and how it might be polished over time. I think you have a great setting and a ton of potential at your fingertips it's just a matter of carving it out. The puzzles are serviceable and some tweaks to the atmosphere or maybe some more visual spice on the maps could do a lot! I always recommend the opening for the anime .hack//sign and Mamoru Hosoda's work on Digimon (particularly Our War Game aka the first major arc of the Digimon Movie) and Summer Wars when it comes to netcore stuff. But that's not necessarily the MS Paint core you're going for and could be more adjacent than helpful.
I hope my thoughts are helpful in fulfilling your vision and good luck in your endeavors! Making games is a pain in the ass!