Really good first try! I'm glad you made it in time! This looks like it was a lot of work to set up, and I'm glad you see your shortcomings. I think the other comment says a lot of good stuff. The only things I have to add are 2 very specific things and 1 general thing:

1. When I was near the vial in the demolitionist's house, I got frozen in place. At first I was afraid the game had crashed, but it looks like it's just triggering that timer without showing the text.
2. Sometimes, especially inside the inn, the beige-white text would overlap the beige-white walls of the building and be hard to read.
3. I am not sure you need a bunch more mechanics to make this an interesting game--just make it engaging to talk to the characters, in part by, like you said, really working on the dialogue system, but also just by making them have a little more personality. I think you have most everything in place for some interesting stuff to happen, and it almost happens, but then it's over. I know it's because you ran out of time, but I am genuinely curious about our conversation with the librarian and about the lower-right-hand house in the town.

Anyway, good job! It was my first time making a game, too, so I'm glad there's someone else here like me. If you make an update I'll probably play it!

Alright, I know this is your first game so you have plenty of room to improve on, so I'll try my best to be as constructive as possible:

First part, from the pure gameplay standpoint.

Aside from the obvious lacking of music and sounds, the walking physics feels bad. Likely because the collisions boxes on the objects are too big, and walking against an object doesn't stop you, but just keep rubbing your character against it causing the whole screen to shake. When you hold down to exit through a door, it takes a second to fade back into the world, at the time where you are still able to walk downwards, which ideally obviously shouldn't happen. The world feels pretty empty too, the background colors make it feels like you are walking on nowhere, and not a single NPC is out on the world. Ideally, you should be able to interact with the world/NPC using a button instead of just walk against it. 

Some nitpicks apart from that would be some small designs issues like the castle not shaping like a castle and doesn't have a throne, the houses feel empty and stuff like that.

Second part, regarding the story.

Well, as a narrative game, the story is obviously lacking because it just boils down to less than 50 lines of dialogue. But ideally, when you are writing a extremely short story, you should not waste a single line just because of chekhov's gun, which most dialogue holds up pretty well, just some of it didn't. For example, the gunpowder line, which was supposed to be a hint to tell that the demoman was the killer, but was never mentioned again outside of the observation itself.

Most Whodunnit games have really linear storyline, but ideally you want to at least let the player make some amount of decisions, even if when they're wrong you just fail them, I can't tell you if my input actually affects the outcome in the game. Which the game has the excellent settings for it, after all, you are just a stranger passing by, who's gonna blame you for putting the wrong person in jail?

Some other nitpicks regarding the plot is that there's some pretty jarring plot holes, such as why would a king live in an inn that's walking distance from his castle, the knight just put his complete trust behind a stranger he just met 5 minutes ago to investigate the death of the king, and demoman just leave his murder weapon lying around, and how the maid from the inn didn't see the murder happening despite standing right outside the door which means she must have known who came in and out of the room, etc, etc.

If you are still interested in making detective games after the jam, I highly recommend checking out this video https://www.youtube.com/watch?v=gwV_mA2cv_0, which discuss a lot of aspects of detective games mechanics