1. Specialized of that era is going to be BSA, but probably the MTB 73mm wide BB shell rather than the old standard/road 68mm BB shell. This BB should be for a 73, but you might want to slip it together and measure - then consult the manufacturer's installation manual online.
2. You probably need the original Shimano external cup wrench. like the Park BBT-19. But count the notches and measure before you buy. Or go into a bike shop that stocks them with one of your cups to try them out.
https://www.parktool.com/en-us/categ...bottom-bracket
3. Yup.
4. You don't remove anything on the drive side. The spindle is a permanent part of the DS arm. You slide the BB body off the spindle, install to proper torque, then install the NDS cup. Lube the spindle and slide it through, then follow the installation instructions to mount the NDS arm, adjust the bearing preload and tighten.
You have done an excellent job of keep it secret what kind of crank you have, so you'll need to look up the specifics as we cannot guess. But this crank appears to install much like many Shimano Hollowgram cranks and BBs.