Should I replace old drywall in closet before I tape?
Nope. It'll finish up just like new drywall would. If the areas with water staining aren't soft they're not a concern, and whether you replace some of it to reduce seams is entirely up to you. Well done butt joints are as invisible as recessed seam joints.
I would consider adding some screws to help prevent nail pops by securing the sheets better, but not too long ago all drywall was installed with nails and it was never really a problem. We use screws now mostly for speed.
Do use a quality sealer/primer when you're done, though to ensure a uniform paint color and texture.