Certain taxes expired on July 22 and are not being collected, including the $3.70 domestic segment fee and 7.5% domestic tax. Others remain in effect, such as the $2.50 per-segment September 11th Security Fee.
Possibly. But Congress might restore the tax retroactive to July 23, in which case you would not. And in any case, the airlines and IRS are still arguing over who should actually issue the refund.
Possibly. Federal law likely wouldn't allow the airlines to collect it from you directly, but the IRS could. This would be a political disaster, but it wouldn't be the first time Congress enacted a deeply unpopular measure.
And now it appears that is exactly what Congress will do tomorrow. (Specifically, the Senate is expected to pass H.R. 2553, which already has been approved by the House.) That means no refunds - and the possibility that if you bought a ticket during the lapse, the IRS will be coming after you for the taxes that you thought you had saved. Swell.