Electrical – the name of a coupling nut with a cutout

First recommendation; put labels on stuff. Sorry, nobody needs a Monday Morning Quarterback.

Do you have a meter? I would try metering out which wire gets voltage with the lightswitch going on and off.

My guess, based on the two way switch, is that the three black wires are; Line ( from the breaker ), feed to the outlet, and feed to the switch. The two whites in the wirenut would be the Neutral to the breaker box, and the outlet. I am guessing that the single white wire is a carrier from the switch, or in your case, switches. With this premise in mind I would check the voltage with a meter on the single white against ground while blinking the switch. If the meter blinks with the switch, then put the fixture black onto this wire, and white with the wirenuted whites.

If this guess is wrong, I have more questions. Did / does the outlet loose power when the lightswitch is off? If yes, then it's probably wired with all blacks together and all whites together. But again, I'd check with a meter before closing in all the boxes.

good luck, let us know the outcome

It is possible, but the energy losses are great, and the insulator (your desk) will heat up, probably causing heat damage. Even then, the receiving coil wouldn't be integrated into the base of the lamp.

In short, there's no way this wouldn't be even more obtrusive than a cord hanging off the side of your desk. If the desk is against a wall, probably better to put in a desk height electrical outlet.

The reason induction charging works for cell phones is due to tiny air gaps, low power requirements, and generally good heat dissipation compared to energy losses. Only a tiny fraction of the used power actually makes it to the cell phone. But the phone is storing that, so eventually (over the course of an evening) the phone is charged. By the way, most phones lack receiving coils, so you still have to plug in a receiving coil and wire into the phone charging jack.

Power that is not directed tends to dissipate according to the inverse square law. Even if it didn't, the losses in transforming the power to wireless and back are quite large. Some of the wireless community is quick to point out the wire transmission losses, but that's nothing compared to wireless.

With a lamp, you'll need a lot more power, and continuous power, so the issues with induction fed systems will become a lot more apparent. I would recommend a battery powered LED desk lamp (with a set of rechargeable batteries). The LEDs will extend the time between recharges.