were they

Yes, it is a rather old fashioned way of using the subjunctive with the verb preceding the personal pronoun/subject. These days many people would write it using 'if':

...66.9% said they would rather be doing some leisure activity if they were not busy working.

I think-
"Activity where they not"
However, when asked what they would like to be doing at the time of the interview, 66.9% said they would rather be doing some leisure activity where they not busy working.

No, that's wrong.

Were is correct. I hear it used in this way quite regularly.

Also 'should they', 'had they', and probably others if I put my mind to it.

They left a key under the doormat, should they ever come back and find themselves locked out in the future.

Their weight loss would have been more, had they stuck rigidly to the diet.

Yes “Were” is correct.
Thanks @Lordyaswas for clear explanation.