I find it confusing with how range seems to be interpreted differently than line of sight for horror check.

Imagine two tiles; one is a hallway and the other is a room. (Both tiles are clear of obstacles or locks)

There's an investigator in the hallway and there is a monster in the room. They are one space apart - and also of course in separate rooms.

The investigator is allowed to fire his/her ranged weapon to hit the monster. (right?)

But the investigator never has to perform a horror check since they are in different rooms. (right?) Isn't that weird?

Yes, this is one weird occurrence; but it only rarely happens. It is necessary for the mechanical simplicity of the game. Maybe the investigator cannot tell that the target is something other than a regular human being. The fact that a monster can incur multiple horror checks over multiple turns by continuing to enter the space an investigator is in can be justified when you consider that the monster is chasing the investigator and the investigator is panicking because of it.

But careful in your specific example: the investigator would not be able to use a ranged weapon because the target would be through a closed door! A better example would be two (contiguous) hallways or outdoor tiles.

Oh! So doors are always closed? Even after the room have been explored? That's great to know. That solves a lot of these situations.

Thanks for pointing this out!

Yes, the hallways and outdoor examples are better

Follow up:

I can kind of see how it makes sense not to get a horror check for a humanoid monster when aiming a ranged weapon or spell at an adjecent hallway or outdoors tile. Your eyesight might be bad, light low or whatever so you're not SURE and you might not see all the scary bits.

But this all goes out the window with beasts and eldritch. If you are aiming at such a monster you know what it is and it should be scary.

Is it common to implement house rules such as this? I kind of feel like doing it

We have been thinking about the same question as well, and usually we just think it's best to go with the rule as written. The investigator might think that he/she has a good chance of escaping if the monster doesn't fall down after the blast from a shotgun since the monster isn't that close of him/her.

A bit off topic, but does everyone play the doors always shut? We usually assume that the doors start closed, but after the rooms have been explored the investigators and humanoid monsters (and Mi-Gos too because they're intelligent) may open and close doors freely as they go unless this option is used to create ridiculous situations.

Doors are always shut. The special ability of the Shotgun is that it can shoot through closed doors.