Depends on your personal preference to some extent. Generally I would use a separate bp.
If you have multiple weapons or if the weapon can be separated from the player, I would give the weapon it’s own bp for sure.
Even if you are only using one weapon you also should consider if you will have just the player to use the weapon or if the enemies use the same weapon. Using a separate bp is nice since if you need to adjust the weapon you only need to do it in one place.
However, if you have only have one weapon then the character bp shouldn’t get too cluttered if you decide to include it. It also may be a bit faster if you’re just prototyping.
Depends on the project and how you want to handle it. There’s not “set in stone” UE law that says you have to do it this way.
If you’re going to be serious about your project, I would most like put in a gun blueprint. For example, my game has multiple guns, so I’ve constructed a base weapon blueprint that my guns inherit from. It handles everything related to guns: meshes, animations, particles, shooting, reloading, playing its sounds, etc. A player just knows it’s an object to pickup and have actions performed when buttons are pushed.