the Triton respirator is OK, we have a couple in the shop, but they have not seen much use as we have dust collection. It does have Ni-Cad or Ni-Mh rechargeable batteries soldered in 3 places inside the fan unit, so when those batteries go bad, it won't be quite as simple as replacing a battery pack. But a Batteries-Plus type place should be able to handle the job.
One other thing to consider about having a shop in the basement is not burning the house down. While egg cartons and carpet scraps may soak up sound, they will also burn. Carpet burns much better on a vertical surface than it does on a floor. Always flame test anything you are considering as sound proofing.
Oh, don't mix wood and metal in the same dust collector, don't mix steel and aluminium dust, and make sure your metal dust collector is designed for metal dust and sparks. A burning filter or "bag explosion" is more excitement than anyone really needs.
There are 4 secrets to soundproofing
1: Don't make the noise in the first place. Put rubber mounts under machinery, spend a few more $ for the quieter air compressor or the precision balanced steel pulleys for the grinder, etc...
2: Air sealing. A hole in a wall the size of your finger will let more sound through than will go through a 4'X8' section of the wall. Seal around pipes and wires, any cracks or gaps, etc. If your air ducts run through the basement, you will have to isolate them, wall them in or sound will just fly through them to the rest of the house. If you screw or nail something to a wall or ceiling that you later remove, patch the hole.
3: Mass, especially mechanically isolated mass. This soaks up low frequencies. You can buy expensive barium loaded vinyl, but drywall is heavy and cheap. Put up a layer as a ceiling, and then add some "sound isolation channel" and another layer of drywall. Be sure to mud and tape the seams in both layers and tape or caulk the edges for the air seal. Take photos or mark where pipes or wires are that you may need to access in the future so you can cut small holes when service is needed. "Access doors/panels" are an option, but they will leak sound.
4: Absorption, this is the fuzzy stuff that soaks up the high frequencies. You can buy fabric covered fiberglass panels or melamine foam that are flame retardant, or go for drop-in ceiling panels/tiles which are fairly cheap and can be tacked or glued to walls and ceilings. They don't burn, just don't breathe the dust you make when you cut them.