Wiring – Add a light switch to an existing switched outlet

There probably is no neutral at the switch box. A switch with "one set of wires" (I am guessing you mean one cable - pictures or a clear description would help me be accurate) is almost certainly a "switch loop." Under newer codes that should be a 3-wire (red-black-white-ground - ground is not counted in naming the cable, so "3-wire") cable to supply a neutral to the switch box, but under older codes a 2-wire black-white-ground cable was often used.

One wire (should be black) will be the hot to the switch. The other (should be red on the 3-wire, or on 2-wire the white wire which SHOULD have red tape or paint applied to it) will be the switched hot, from the switch, back to the first fixture. The neutral for the circuit will be at the fixture in this case.

(Pardon - North American color codes - other parts of the world do have other conventions, involving blue, brown, etc...)

Likely what you need to do is move the new light's wire to one of the other lights, not the switch.

Edit, Add - what you probably are doing is putting the one new light in series with all the other lights in parallel between it and neutral. If you remove the bulbs in all the other lights, odds are good that the new one will go out. If you remove most of them, you'll probably find that the new fixture gets dimmer, and the remaining lights on the circuit start to glow a bit...all the lights should be in parallel (from hot to neutral) but what you have now is hot, the one light, an intermediate connection from the one light to all the other lights, and through those other lights to neutral.

Yes, this should work. A few things you may have already considered:

1. Each time your cable goes into or out of a junction box you should plan to lose 6" in length.

   So tying into and out of the ceiling box for the fan would require 12" of slack on the existing 4-wire cable. Your plan to add a separate junction box is a better approach. The trick is to put it at least 6" closer to the dual-wired receptacle and run 3-wire the longer distance to this box. When you tie into the 4-wire here you will have extra 4-wire to use in the ceiling fan box.

2. Don't disconnect the red wire going to the duplex receptacle.

   Just tie red and black to the un-switched black in your junction box. This puts half the load on each wire and you don't have to rewire the receptacle in the wall.

3. Get a deep 2-gang switch box to fit all those connectors

   and dissipate the heat any dimmers you use to control fan speed or dim the fan lights. You're looking at needing at more than 29 cubic inches not counting extra for dimmers. And you always want to leave room for future expansion.

4. Get a ceiling box designed to hold a fan and mount it securely to a joist.