Electrical – Is this an appropriate way to add a single-pole switch (without a gang box) to a circuit with an existing single-pole switch

It entirely depends on how the existing switch is wired. You need a constant hot and a neutral wire. It would be helpful if you can check what wires are in each location - what colors, and how many actual cables are coming in. You'll need to turn off the power at the breaker panel and physically remove the switches (don't disconnect them, just pull them out), as well as the light fixture. Compare to the diagrams below (or at the original source), or ask here again and we'll try and help you out (labelled pictures == very helpful).

There are several ways to wire 3-way switches, but what you'd be concerned with is the type where power comes to the switch first:

In the above diagram ,you can add another switch to the one on the left - you'd connect to the white and black wires coming from the "power source" line.

On the switch on the right, you have a neutral, but you don't have constant hot -- hot is switched, on either the red or black, depending on how the first switch is flipped.

Another common way the switches are wired is the power goes to the fixture first.

In this case, there is no neutral at the switches. (Also note, the black tape on the white wires indicates this. White is, by code, always neutral, and has to be marked if it's used for a switched circuit. That said -- keep in mind that not everyone follows code.)

If you're missing the hot/neutral, the only option you have is to run a totally new wire, or possibly (depending on how it's wired), you can convert the 3-way to a single switch, and then re-purpose the wires to supply constant hot and neutral to the new outside switch (effectively, one of your 3 way switches would get converted to control the outside light instead). This still may not be possible, and definitely isn't a beginner task - you need to understand electrical fairly well, and map out everything involved with this circuit.

Something is wrong with the wiring to the ceiling fan or with the connections to the switch.

Possibly in between the last succesful test and the first unsuccessful use of that switch something has damaged the wiring (e.g. a nail into the wall? strain on the wires?)

I would isolate the circuit at the breaker box, test for voltage at the switch and then test the wires attached to the fan switch for unexpected continuity between live & neutral, live & earth. I'd then detach the switch from all wiring and test for continuity across it's contacts with the switch in each state.