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.
It's definitely a interpretation issue with the local inspectors. You can improve the installation in the picture by following the guidelines in 110.26 they may allow the wire within the working space about the panel to be not subject to damage but the water line should be rerouted to be outside the working space described in 110.26 also the wire should be supported per 334.30.
Best Answer
Assuming:
No. It is not the slightest bit dangerous.
Aesthetically and functionally it would be nicer to have the wires, and that beam, out of the way to have a more open shower. But the electrical safety is not a problem at all.