Can I use a saucepan to cook non-liquid things, like making omelets?
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.