Move the run over a few feet, which would it stick it beside the air duct (I think there's about 1/2" between the duct and the adjacent stud. Is it safe for them to be so close?)
There is nothing I see in the NEC that puts restrictions on running electrical line near HVAC as long as it meets other code requirements. It must be properly anchored to the structure, meaning that when it is run it should be anchored or stapled to the wall studs. HVAC does not count. In absolutely NO circumstances should you ever run electric line INSIDE of HVAC!
Leave the run as is, move Lord Duckington from his throne, cut away the plaster between the two hole, drill holes in the studs and feed said wire through it, fix mess (I think this wall is load bearing, so this might be a bad idea).
A small hole for electrical line in the middle of the stud will not affect the load bearing capacity of the wall by any meaningful amount. This is safe and probably the best option.
Re-wire so that this run is getting its power from the circuit on the wall behind it. They don't have much on them, so I think combining them is probably safe. In this case, would it be safe to have a few inches of romex going from one box to the next and just connect everything with wire nuts?
This is another option and would probably involve less plaster patching when you are done. The only thing you should make sure of is if the lead line is coming to the switch box. If the power is coming to the receptacle or luminaire rather than the switch then this will not work. Make sure to wire everything properly in parallel if you do this.
Pull off the baseboards, drill a new hole in the floor, and move the box a 6 inches to the left (not sure if the lath is behind the baseboards or not).
If you feel that there is enough room there not taken up by the HVAC system then sure, you could rerun it this way from the basement. In fact, with your tall baseboard you can hide the plaster damage and fishtape it up to the new outlet location. Again, please make sure that there isn't a vent running up that wall stud going to upstairs, which seems likely from your photo.
Move
Lord Duckington I am sure doesn't approve.
Almost definitely no.
I'm not familiar with that exact model, but typically non-contact voltage detectors are designed to have to be within a few mm of the wire to work. They're called "non-contact", but that means no electrical contact (i.e. they don't have to touch the bare wire). You'll just about have to touch the tool to the insulation of a wire for them to work properly, so it's not likely you'll be detecting wiring within any type of wall with one of these tools.
The description from the Fluke website for the Fluke 1AC II A1 VoltAlertTM Electrical Tester, seems to agree.
The next generation VoltAlert™ AC non-contact voltage testers from Fluke are easy to use – just touch the tip to a terminal strip, outlet, or supply cord...
Notice that it says, "just touch the tip to...".
Best Answer
Here is an easy method to determine the location of live wires, with a stud finder. I have a stud finder on which there is no means to adjust the sensitivity of the live wire detector. However, you may use paper for the same purpose. Stud finder is sensitive to the electric field; electric field depends on the distance (i.e. how far you are from the live wire) and the medium (i.e. the material between you and wire (air or other material). If you put some paper between the stud finder and the wall you decrease the sensitivity of the stud finder. I use phone book pages for this purpose (they are free and you may add as many pages as needed). The thickness of the pages may go from one page to 1 centimeter or more depending on the current in the wire, and the paint used on the wall. You will see that the stud finder sounds for electricity only when it comes on top of the wire; you can pinpoint the location of the wire exactly. I hope this info helps someone. Good luck.