What is the minimum thickness of foam board when insulating a basement
Building codes will tell you this information. First, find your climate zone on this map:


Then cross-reference that with the table here: http://publicecodes.cyberregs.com/icod/irc/2012/icod_irc_2012_11_sec002.htm

Look at the first number under "BASEMENT WALL R-VALUE". That's the minimum R-value you need to be safe. Divide that number by the per-inch R-value of the foam you want. For example, if you live in zone 4, it says you need a minimum of R-10 on the walls. If you're using low-density Home-Depot-sourced R-3.8 EPS foam (a sheet of white beads, like styrofoam), that means you need 2.7 inches of EPS  to be code-legal and safe. Round up to 3 inches. XPS (blue or purple and smooth) is R-5, so you could get away with only two inches of XPS. But XPS is more expensive per R-value, worse for the environment, and degrades to R-4.2 over time, so in general, EPS is preferred. Seal the edges of the boards with little bits of spray foam, caulk, and tape (as appropriate). Seal them to the floor too. You can use the same method to insulate the rim joists. Just make sure everything's sealed up tight so no inside air can reach the walls and rim joists. 

The product you reference is 1/4" thick and only rated R-1 per board. So you would need ten sheets of this product to reach R-10. It would be much more cost-effective to buy a single 2 or 3" board. I believe Home Depot sells 1-3" sheets of both EPS and XPS so you should be able to find what you need there. You can use multiple layers, and in fact this is slightly preferred because you can stagger the boards' seams.

With continuous insulation like this, you don't even need to frame an interior wall, which will save you time and money. Just install drywall right over the insulation using Tapcon screws going into the concrete wall behind the insulation. I recommend using cementboard instead of drywall for the bottom few feet for flood resistance. If you insist on framing a wooden wall over the foam, use pressure-treated lumber.

There is no issue with leaving the insulation exposed for a while besides fire safety. Polystyrene foam (which is what both XPS and EPS are made of) will ignite and melt at relatively low temperatures. You are required to cover them with an ignition barrier like drywall. You can wait to do this as long as you're comfortable with, or as long as your inspector allows you to, if this is an inspected project.