Chis is right in what he is saying. To further elaborate, I assume the structure has a full raftered roof. You need to be absolutely sure if any portion of the wall you want to remove is a supporting part. It is rare that a partition wall running parallel with the ceiling joists is a supporting wall, unless it spans across the center axis of the house perpendicular to the ridge board and some type of post or truss comes from a rafter to the top of that wall. When you look in the attic, are there any angle trusses attaching to the joists, most likely in the center of the building? Is there a center wall on the first floor that runs the length of the house? This is usually indicates a bearing wall. If your wall in question runs parallel with the rafters and joists it most likely is not supporting, especially if there is no joist above it. Since you will have to repair the ceiling after you remove the wall, you will need something to attach your sheet rock to, unless the existing ceiling runs the length and the upper plates of the wall you are removing are located below this ceiling. In any case, you will want to install a joist. If the removed wall is not supporting, it really doesn't matter if you install before or after. other considerations are if there is any electrical or plumbing in the wall. you gotta plan for that as well.
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.
Best Answer
Each truss manufacturer have specific requirements for drilling holes in their joists. You’ll need to verify the JOIST TYPE (structural type based on flange size, species, etc.) and SPAN with the manufacturer and then determine where they allow holes.
However, here is an installation guide
https://www.bc.com/wp-content/uploads/2015/02/UK_Install_Guide_A2.pdf
from Boise-Cascade which I’m sure will be similar to most manufacturer’s installation instructions. If you notice in the upper left hand corner, holes are to be drilled in the center (the neutral axis) and size depends on distance from the support.
These joists are specifically manufactured based on loads, spans, etc. and modifications to the joists should be done to the manufacturer’s instructions....random assumptions about drilling 1 1/4” holes just 1” from the bottom flange is not recommended.