It depends on the type of mount. If it's an articulating arm mount, the bending moment at the wall is going to be a lot higher than if you're doing a flush mount. In that case the safest option would be to open up the drywall and put 2x4s inside the metal studs, and use the included wood screws to attach the mount to the new 2x4s.
If this is a load bearing wall, and you're certain the studs are heavier 20-gauge load bearing studs, you might get by mounting an articulating arm using toggle bolts like these. If the wall is non-load-bearing, it's likely made of lighter 25-gauge studs, which just don't have the localized rigidity to support this type of mount so you're stuck reinforcing with 2x4s as I described above.
If you're doing a flush (or nearly so) mount, then you can get by with toggle bolts even if you have the lighter studs.
I wouldn't recommend any kind of self-tapping screw in a load bearing application like this. It doesn't take much overtorquing (especially with 25-ga studs) to leave them just on the verge of pulling right back out.
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
If the steel stud is of sufficient thickness/strength you could fasten it with self-drilling Tek screws. I would personally be more concerned with the weight bearing capability of the steel stud (assuming it is the lightweight steel studs commonly used in commercial build-outs in my area).
Most ceiling fixtures (you use the term stud rather than joist so I am assuming wall mount) that are expected to hold any weight should be attached to a cross-brace which is attached to two joists.
Consider using a long pilot drill and drill through both the front and back sides of the steel stud, then use a long self-tapping screw to penetrate both sides of the stud.