I did a lot of research when finishing our basement. I eventually went with a wall model recommended by building sciences corporation that, from outside in, is:

• existing exterior wall (concrete, concrete block, etc)
• foam board insulation (XPS or EPS, I went with EPS)
• stud wall
• sheet rock (I went with a paperless product called Densarmor

This is a system that Fine Homebuilding magazine also recommends and is considered valid by the US Department of Energy. I know this because our local building codes were still using the antiquated 'fiberglass + plastic' model and I had to do a ton of research to educate the local code enforcers before they'd approve this.

I went with metal studs for the following reasons:

• they're all perfectly straight
• I can carry 20 of them at a time (makes it REALLY easy to haul into a basement)
• easy to build in-place (no need to frame then tilt-up walls)
• cut with tin-snips in a matter of seconds
• mold can't grow on it
• can be installed without screws (can be crimped in place)
• at the time, were the same cost
• wiring channels are built-in
• you can use thinner studs (it's impossible to find straight 2x2's in wood around here)

There are a few cons, though:

• you can't easily nail into them for attaching baseboard
• you still need to frame out your doors with wood for the added strength
• you can't mount cabinets to the wall with metal studs

As for baseboard, I decided to use the new synthetic foam pre-finished trim. It looks pretty good, is super light, easy to work with and...it's not wood. So I thought it was another great product for a basement. Because it's so light, it was really easy to toe-nail it in to the sheetrock with an pneumatic trimmer.

As for mounting cabinets and such, on the walls where I knew I wanted to do this, I added 2x2's inside the metal studs for support.

The only corrosion issue that I'd be worried about is rust, and that should only be an issue if you still have a moisture issue in your basement. It'd also take a really long time for a stud to rust through and be any sort of problem.

Some tips:

• be sure to separate the floor plate from the concrete. I used 1/4 XPS for that and then power-actuated hammered them into the concrete. This thermal break will prevent moisture coming in through the concrete to condense on the metal
• don't screw them in. I did and while it's not that big of a deal, they make crimpers just for this purpose. Invest in the crimpers as it'll make things go really fast.
• be sure to buy plastic grommets for the electrical channels. You don't want your electrical cables rubbing up against the bare steel edges.
• wear really good gloves

As for your plan:

barrier (tar paper) on all exterior walls, Framing a stud wall, insulating, installing a vapour barrier, and dry-walling over top

...I STRONGLY recommend against that.

for starters, your plan involves two vapor barriers...that is a really bad idea. That will only trap moisture inside the walls. The modern recommendation (at least in colder climates) is to not use any vapor barrier in an old basement. Instead, use foam board for the insulation. Foam board is permeable, and the idea is that if water ever got on one side or the other, it could eventually dry to the other.

The other issue is that you want the insulation on the OUTSIDE of the stud wall. The foundation wall will be the coldest surface and is where moisture would condense. You want all of your framing on the inside of the conditioned space.

The proper way to put a water barrier in a basement is on the OUTSIDE of the foundation. Ideally, you'd have a water barrier and insulation on the outside of the concrete. But that's obviously really hard to retrofit.

Right stud: What's the best way to screw the stud into the frame at the top and bottom so that it's sturdy enough to support the new wall I am going to hang on it?

What you're doing here is called "sistering". I personally would use 2 screws each at the top and bottom. More is unnecessary - after all look at the other studs - nothing holds them in the middle.

Left stud: Since I will be able to screw this stud into another stud, how many screws should I use? Should I put the screws in pairs so that it's two screws side by side all the way down?

For the left stud you'll need to toenail - that is - install the screws diagonally into the top and bottom plates. It'll be easier if you pre-drill the holes - install at a 45 deg angle, 2 top 2 bottom again.

What length and type of screws should I use for both studs? I am assuming wood screws for this job, but I am not sure about the small details.

2x4's are actually 1.5 by 3.5. So you'll be sistering into a 3" depth, so use 2.5" screws for the sistering, and on the diagonal, use 3.5" screws to cover the length of the hypotenuse and give you good bite w/o over sinking.