Electrical – Checklist for installing proper electrical outlet outside the house

IMO it makes absolutely NO sense to use an existing circuit, especially considering how taxed they are. As long as your panel will accept tandem breakers what is the issue? Also, with a house of this vintage I highly doubt the wiring is grounded, which you cannot extend. And do not assume if the wiring is old AC/MC cable that it is grounded, much of it from that vintage is not, regardless of the metallic sheathing. The only metallic wiring like this that can be used as a grounding conductor is that with the think aluminum bonding strip running along side the insulated conductors. Besides, you never know what you might use this new receptacle(s) for. Don't think you'll never use it for anything more than a laptop charger.

Also, you re-purposed the old A/C circuit for a dryer?? Was it at least 10/3 cable? If not you have a non-complaint and potentially dangerous circuit if used for a 120/240V electric dryer.

Your current plan is no good -- first off, 400.8 point 1 forbids the use of cords as a replacement for permanent wiring (stuffing a cord down a conduit certainly counts, and is also prohibited explicitly by 400.8 point 6). Second, 400.9 prohibits the splicing of cord during installation. Third, wire splices need to be in a junction box so that they can be serviced in the future.

A better plan would be to bring the conduit out to the receptacle box where you plan to tap power, and then stuffing NM through the conduit (yes, this is OK). At the box end, you'd simply tie it into the rest of the wiring as you would ordinarily (using pigtails and wire nuts if they aren't already present there). At the monitor end, you'd attach a field-fittable IEC C13 connector directly to the NM cable and plug it into the monitor (you may need a right angle connector here -- make sure it accepts 14AWG as not all do).

Your other option is to use a recessed receptacle in a junction box mounted at the monitor end of the conduit, and plug the monitor's cord into it. This requires that the cord either be exposed completely, or the monitor be readily removable (i.e. "take a few fasteners off, remove a trim piece, and it comes out", not "oh, we have to tear into the wall to replace this") in order to apply 400.7(A) point 8. It also requires a way to get the cord to the recessed receptacle, considering you can't bury the junction box + receptacle in the wall.

Given that you are dealing with an AC adapter -- the recessed receptacle approach is the superior one (I don't believe you can get the smaller IEC connectors to fit on 14AWG anyway!). The main issue will be finding a pairing of recessed receptacle and shallow box that will fit together while accepting the PVC conduit. I'd start with a P&S P108W as that box is only 1.125" deep, leaving 0.375" for the cable and faceplate, and see what recessed receptacles work with it -- the silly folks at Leviton don't publish a dimensional drawing for the 689-W linked above, so I can't tell from here if it fits or not. (Drilling a 1/2" conduit KO in the above box should be OK btw -- the FS-type boxes that are designed to fit conduit are all too deep for this job.)

Oh, and if you do use the recessed receptacle approach, you'll probably need to finish out the monitor-alcove with some half-inch drywall back-to-back with that plywood backplate, using screws and standoffs to mount the electrical box. (Boxes that stick out from their surroundings are kosher by Code.) Of course, the only way any of this will work is if the cavity in the monitor will fit all of the cabling as well as the transformer and the J-box.