If it's an old house with good concrete foundation walls and good drainage, and hasn't leaked yet, you should be safe.

Another point - be sure that the basement wall vapour barrier plastic is sealed to the insulation's vapour barrier in between the joists against the exterior wall. (back in the 50's or before, vapour barriers were an occasional concept, usually that paper-backed fiber insulation was the vapour barrier. Proper insulation was also often just a suggestion.)

Also, by code - while pink fiberglas insulation can be exposed (with a plastic vapour barrier) - Styrofoam insulation must be covered with drywall; otherwise, a fire could burn the Styrofoam to produce noxious fumes. Sealing it behind drywall limits this risk. Also, know your code - electrical outlets in a finished room must be no less than 12 feet apart (since many appliances have 6 foot cords) not counting doors and windows. My local code allows for zero outlets in unfinished areas. (i.e. no drywall).

And... finishing a basement (or part) can add to your tax bill if the municipality gets wind of it.

I thought Robertson (the square hole screw heads) was more of a Canada thing?

Yes, my go to reference for construction, Holmes on Homes, says NEVER have bare untreated wood touching cement. It can absorb water from the cement.

You also have to consider the risk of basement leakage. Cracked foundation wall leaking, or rain/melt water buildup leaking in, may damage the bottom of your walls. Is this a risk? How good is the foundation drainage? Would you notice?

And, in much older houses, the concrete basement floor may not be particularly level. A raised wood subfloor allows you to level things so you can use tiles or carpet. But you may have a central floor drain, you may never notice if there's a water seepage until mold shows up, etc. Basement floods do more damage if you have to replace a raised floor. (I saw a Holmes episode where he used a floor leveler compound, like a very wet cement that leveled like water but hardened - to get rid of excessive slope. Don't block the drain. )

Exactly - why would you want air movement behind the interior walls? All it does is guarantee that if there's one cold spot, instead it makes the whole area behind the insulation a cold spot. Especially the ground level cold will migrate to the basement floor level, spreading the love. Insulation flat against the walls makes sure any cold does not spread beyond where it's cold through the wall.

In Canada, where winters are probably colder at times, the standard now is 2x6 exterior. Or... for basements, a sheet of styrofoam insulation, then insulated 2x4. If you plan to put weight on the walls as others mention, with cupboards, shelving, or tool pegboards- you want the strength of 2x4 on standard 16-inch centers.

The vapour barrier should be on the hot side of the insulation, so inside humidity in winter does not migrate inside the walls to condense. Usually for pink fiberglas insulation, this is plastic sheet on top of the wall, stapled to the studs. Typically, it's "glued" using acoustic sealant so the holes from the stapes are sealed. Electrical outlets 9and cable TV, network, etc.) are on the warm side of the plastic, using tape to seal the wire coming through the plastic. the key is NO AIRHOLES.

(Alternative construction is sprayed in expanding foam, which is then sawed flat level with the studs when it hardens. Since it's sprayed as a liquid and forms a continuous piece, a vapor barrier is unnecessary, it's already airtight.)

Take pictures as you do the work, especially of the walls before sealed from view. Not sure where you are, but here you can do electrical yourself if (a) you know what you are doing and (b) with the supervision of a qualified electrician. (I.e. he/she says "yes, that's up to code".)

Some municipalities require a building permit, especially if you touch electrical or plumbing. I has a relative in the USA, when she sold her house 20 years later, someone somewhere went through the house's building permit history (house inspector?) and claimed the family room behind the garage was never issued a permit to be finished. AFAIK she bought it finished. Then bought and remortgaged when she bought the half it from her (ex)husband, with no hassles. I sold my house in Canada with no such hassles. How persnickety is your municipality?