Can I use a saucepan to cook non-liquid things, like making omelets? I know that saucepans are good for cooking things in a liquid or with decent liquid content, but are they good for sauteing or making omelets, just things that don't involve little liquid?
Yes you can.

BUT.

Saucepans have high walls and will make handling something like an omelet more difficult; spatulas will have a hard time getting under the omelet.

The small surface area compared to a frying pan will not make it perfect if wanting to sautée vegetables (for example); if you put too many ingredients in the pan, then it will steam instead of browning.