We have plenty of cat5 in the house.
Is it OK to use one cable as stereo audio (connect headset with TV)?
Yes it is as long as you send the L-R signal over specific pairs rather than any arbitrary four wires - the twists in the pairs are tightly controlled and very effective at rejecting noise.
I have heard tale of audio being run successfully over 300m of cat 5 although for this distance you really need to use balancing transformers - so-called Baluns - eg:
They're not cheap (cheaper ones are available), but under some circumstances, using baluns and Cat 5 can be less expensive than running dedicated audio cable.
Just to confirm Linker3000 answer, this is indeed possible. Keeping to the matching pairs is the key. I've built multi-studio radio stations using CAT5e without problems (and at greatly reduced costs from "old fashioned" multi pair shielded pair cable)- in those instances using a line of products from StudioHub, though there are several other high quality producers of this line of wiring gear.