Electrical – Prevent drywallers from cutting wires

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.

There are only 2 correct and safe ways to add a 3-prong outlet where formerly there was only a 2-prong one*:

• EDIT: now allowed in Code since this answer was written: Retrofit just a ground wire, using proper wiring methods. There is no need for the ground wire to follow the same route as the power cable. Grounds can be shared: You don't need to homerun back to the panel if you can reach another grounded circuit with the same size or larger ground wire also coming out of that panel.
• Safe way #1: run a new cable with an appropriately sized ground all the way back to the breaker panel.
• Safe way #2: Use a GFCI outlet (with the "test" and "reset" buttons), which will cut off power to the outlet if it senses an imbalance of current. (EDIT: As @EricLippert correctly points out in the comments, you can also replace the circuit breaker with a GFCI breaker instead of using a GFCI outlet.)

There are many wrong ways to add a 3-prong outlet, most of which are dangerous and some of which are hard to spot or test for:

• Wrong way #1: Using a 2-prong to 3-prong adapter without properly grounding the tab (common but obvious).
• Wrong way #2: Replace the 2-prong outlet with a 3-prong one, but only hook up the black and white wires (this is hard to see unless you open up the outlet box, but an outlet tester with reveal the problem).
• Wrong way #3: Replace the 2-prong outlet with a 3-prong one and short together the neutral and ground wires (this is probably the sneakiest because it will test as OK with an outlet tester but is just as unsafe as #1 and #2).
• Wrong way #4: Replace the outlet with a 3-prong one and connect the ground to a random piece of metal (the outlet box, a pipe, metal cladding on an armored cable). This too may check out as OK with an outlet tester but is not safe.

*There are actually a few other ways to ground an outlet safely, e.g. with armored cable with a bonding strip or a grounded box plus an outlet that allows grounding through the strap. But those are both pretty uncommon.