4. Anything else I should be aware of before going into this? I feel the Supernatural universe is a really rich one, with a lot of room for expansion, but also room for me to screw up.
To begin with, you should be aware of the legal issues: you can't just use an existing setting without permission without some pretty serious legal risk, even if the game you're making is for free. If your game comes to the attention of whoever owns the rights you might receive a cease and desist letter requiring you to scrap your project, or you may even be taken to court.
You could certainly make a game featuring monsters, demons, and monster-hunters, but you would be much safer making up your own rather than using the existing setting from Supernatural; you certainly shouldn't use the specific characters or actual story-lines unless you're really happy to take a huge risk.
1. What are the issues with the Supernatural Narrative that makes game development difficult for it? I feel the story suffers from overpowering their characters. In the first series demons were dangerous (Still manageable, but dangerous.), now they kill so many it feels like they are cannon fodder.
//EDIT: Sorry, I somehow read this completely incorrectly, I think you were actually saying the same thing as me...
Actually I felt the opposite to this -- in the first series most monsters were tough but beatable whilst demons were nearly unstoppable -- until the Colt was retrieved there was no known way of killing them, and after it's introduction it was still thought to be the only way for a long time, and devil's traps and exorcisms didn't seem well known. Apart from some major characters they actually seem much weaker overall with numerous things that can kill or incapacitate them. They can now be killed with the colt, the knife, the first blade, an angel blade, be smote by angels, be killed by leviathan, be controlled and killed by "special humans" who are tainted with demon blood, be destroyed by salting and burning their original human bones, can be kept out of a person with a ward, can be locked into a person with a ward, can be hidden from with hex bags, and probably numerous other weaknesses that have gradually cropped up.
...and there lies what I think is probably the biggest weakness of the setting; I don't think it was originally intended to last nearly so long, and as it has grown over time a lot of stuff has been tacked on, leaving a setting that feels a lot less coherent than when the show first started.
2. With the above in mind what setting should the game be based? Should I include the characters from the TV series or shall I focus on my own character? Do I focus on some story arc involving demons or more on the general monster hunting?
Obviously given my original advice I think you should focus on your own characters, and even your own similar setting rather than directly using anything from Supernatural.
My wife and I enjoyed some of the earlier seasons of the show but have been finding the larger story arcs a bit over-the-top; we really like the episodes that are general monster hunting and wish there were still more of them. You could potentially have a bit of both by having a major story-arc that carries throughout the game whilst the player is kept busy with various monster hunts.
3. What perspective? I feel a first person perspective works for the horror genre of the TV series. (Hasn’t been scary since the third series.)
First person can indeed be a good choice for horror if you're actually going for scares, but as DiegoSLTS mentions the show is often more about light entertainment and comedy with the occasional scare rather than a proper horror. You might also consider how you'll best represent all of the tasks a player might go through when solving mysteries and hunting monsters; this might be hard to present in first person. I actually think something like an adventure game might be really fun.
Hope that helps!