Carriers will go to sea quite often without an Airwing embarked. After overhaul work, they will do some sea-trials to make sure they have all the bugs out of the systems.
Once the Strike Group starts workups, the Airwing will embark in various amounts, depending on where in the workup cycle the Strike Group is, for example the first exercies might only have abouy 60% of the aircraft, but all the personnel embarked.
When carriers go out for CQ (VT or FRS) then it will usally have 3-5 helos on board. These will do the SAR for the CQ period. Fleet squadron get the extra fun of doing these at-sea periods. I would say the majority of CQ periods have been covered by the Airwing HS/HSC squadrons, but some of the expeditionary HC/HSC squadrons have covered periods over the years.
Pretty much any time a carrier goes to sea the CO wants to have at least two helos (one to fly and one as a back-up) embarked. It makes life a lot easier for the ship if a man overboard happens, embarking or returning the harbor pilot and for anything else that may come up like logisitcs missions. I've been on carriers that have gone to sea without a helo embarked, but since at-sea time is now at a premium due to compressed turnaround cycles, if a carrier goes to sea, it will be covering some requirement that will require a helo on board.