Electrical – use old abandoned iron pipe as electrical conduit

The flexible armored wire you are seeing is often referred to as AC or BX cable. (BX was a brand name like Romex is for NM) Bx was used extensively prior to 1960ish and is still used today. There are no current code requirements that require this kind of cable in residential construction and it has almost all but been replaced with NM. As you have probably already figured out, BX uses different box connectors, and many of the device boxes you have in place were probably made to terminate BX cable.

Although it is perfectly legal to use, and actually quite safe, it is rarely used because of it's higher cost and relative difficulty to install and terminate. The only caution I would emphasize is the ground conductor in older BX. it was very common not to use a separate ground conductor when using BX, and often when there was a separate ground wire, it was a smaller AWG that the black/white/red current carrying conductors. this was never a problem if the BX armor was properly attached to the metal junction boxes and again properly connected at the panel, thus making a substantial grounding path.

As far as the bonus question. BX is heavy and probably brings a good price at the metal recycling brokers. A couple of years ago when junk metal was really hot, a good load would have been worth hundreds of dollars. It was a popular find for junk dealers and the target of theft.

It looks like the Green or Green/White wire is also hooked up. He used 4 twisted pairs, most commonly used as Ethernet wire (the one that you use for internet on your computer, router, modem, etc.) and is most likely a Category 5 cable. It is a relative cheap cable that has 8 wires, looks like he only needed a few but it is easier and cheaper to use what you have sometimes.

As long as your system works and the wires that are connected aren't getting hot (meaning the wires are too small for the electric load going through them) then you're fine. The connectors to the attached wires look like waterproof ones and sounds like you can trust the contractor on that one.

I'd wait a while to make sure everything works, or ask them to do it if they're coming back already, and cut the unused wires close to/at the sheathing (blue covering in your example) and put some tape around that to ensure you don't trap some water in there.

The unused/exposed wires left alone can/will have some copper corrosion form but they are unused so it's more of an aesthetic concern which you can address.