>How close would an Earth-like planet need to be to detect our signals, and what signals would be easiest to detect at this distance?

The wave front of our radio broadcasts are around 100 light years away. TV broadcasts are just radio signals. With a large enough radio telescope, someone could in principle detect those signals. The signal attenuation over that distance would make it very difficult to seperate signal from background noise.

>How does this distance compare to the observable universe, and what does this mean for the search for extraterrestrial life?

100 light years / 13.8 billion light years. Identifying a technosignature would answer the "are we alone in the universe" question but the further away the origin of the signal, the less chance of engaging in meaninfgul dialogue with any discovered species.

>Is it theoretically possible that there are signals from developed civilizations that we haven't detected yet because it hasn't reached us here on earth, because of speed of light limitations?

Yes. There also may be centuries millenia worth of signals we didn't detect because we weren't even capable of looking for them more than 100 years ago.