Electrical – Why would turning off a ceiling fan via a wall switch causes 3 seconds of TV signal loss

Depends on how it is wired. The single switch will control the fan and light simultaneously. If you want the light off, then you pull the chain for the light. The nice part about having the light on a different switch is you can simply go into a room without having to turn on the fan every time and have a nice bright fixture in the room. The downside is that you have to hit the switch every time since the pull chain kind of loses its purpose and is now redundant.

Now, whether you have three speeds or not is up to the manufacturer of the fan. Most fans made recently have three speeds. We have to know the make/model to find out for sure. Doubt it is broken though.

It sounds like you're poking around in the wrong hole. The circuit you describe looks something like this...

Which obviously is not possible, since you say the light is on. More likely, the switch does not control the light at all, and the circuit looks more like this...

It's possible (as previously mentioned) that the switch actually controls a receptacle, or some other device. In which case the ceiling box is probably wired directly to the circuit, and is intended for a fixture with a built-in switch (pull chain).