Easy way to fix separated corner bead
dont use screws on corner bead, that's what nails are for.
after nailing run a piece of fiber glass mesh tape up the crack, but not too close to the corner.
take some painters tape and run it up the molding. when running your mud over the tape, float over the entire area between the door molding and the corner. Bumping you knife up against the smooth surface of the molding will eliminate alot of the rippling in the mud due to the texture.
as for the texture...it will be hard to match up unless you use a hopper. the can sprays you can buy at hd or lowes do not work for the depth of that texture.
I'd suggest if you do decide to go with the can textures to run your mud to very top of the door mold. In doing this you hide the transition from new to old by keeping it above eye level. Good luck