I'm starting to look at TES V Edit for cleaning Skyrim mods and mergeing ESP's. I am trying to install as little as possible on my main PC. Can I use my junky pc to clean mods and merge my mods and their ESP's (mainly just to gain extra ESP slots) then just throw them on a portable harddrive and install them with NMM or Mod Organizer on my clean PC?

Yes you could but TES5Edit is a small program that doesn't need to be "installed" so it's probably not worth it. Just unpack the TES5Edit into a folder of your choice and run it. I have a Tools folder where I install TES5Edit and BSAopt for manipulating the ESP and BSA files.