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.
Well, your pictures have twigged an occasional issue I run into where I can't see them, (likely not your fault) but flying blind....
From a functional point of view you really don't have to worry about separation. Twisted pair is actually quite good at ignoring noise, and 60 Hz noise is of little note to 100MHz ethernet anyway. You can do it all wrong and it will work, 99% of the time. I don't suggest that you do it all wrong, I do suggest that you don't freak out about it.
From a code (and safety) point of view you should not have low and line voltage going through the same hole, and from a hyper-cautious network standpoint they should be separated by a good 12" when parallel, or cross at 90 degrees if they need to cross. Separation matters a lot more in an industrial environment with noisy devices on the powerlines than in a typical single-family residence.
From a "best practices" point of view, network and power in separate stud bays (when running vertical) or between different sets of joists when running parallel to the joists is certainly a best practice, though not required by code - it maintains separation quite aggressively.
Notches are far worse than "holes in the center third" (top to bottom) of a beam, joist, or header. Holes should not be too close to the ends, even in that center third. I'm not sure there's any need for the cables to go through the header if they can "float by" (remember, I'm flying blind here)
Portable electric heaters are nasty, nasty loads. If you are plugging one into a 15 amp (14 Ga) circuit that, all by itself, with nothing else on, is loading the circuit very near to maximum (if it's a 1500w heater, actually more than is permitted for fixed/hardwired loads that might (as with a heater) be on for more than 3 hours. A warm wire is fully expected under those conditions. A 20 amp (12Ga) circuit would be more appropriate if you are in the habit of using one.
Best Answer
Options abound, but I like side cutters or tile nippers (similar to 'end cutting pliers'). With either, grab the leg of the staple where it enters the wood and pry it up. Don't ever pry against the cable.