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I think the wording of the official announcement is unclear. There will be a transition period as the F-35 enters service and the 2 carriers migrate from being entirely helicopter-equipped to true mixed-platform multi-role vessels. In that case, one carrier will re-equip ahead of the second based on operational commitments, refit cycles etc. However, I do not believe the announcement is intended to mean that only one carrier will be fitted with catapults and F-35s because that's not a sustainable capability. The UK would lose it's afloat F-35 capability when that one vessel was in refit and the whole rationale for 2 carriers was to sustain operational effectiveness taking account of periods when the vessels need to be in port.
The UK's military doesn't make a substantial contribution to US military deployments today - witness the relative ease with which the US moved into areas of Afghanistan that had previously been the bailiwick of the British Army. The purpose of a carrier is force projection without the need to rely on host-nation support, and there are lots of use-cases for that type of operation - non-combatant evacuation operations, special forces (eg Sierra Leone in recent history), anti-piracy or counter-drug ops etc.
Apparently. According to AvWeek the UK will now field 1 conventional carrier with F-35Cs.