Aircraft carriers provided strike aircraft for Bosnia, the Gulf Wars and Afghanistan.
Yes, albeit not very effectively. For all these conflicts carriers were important but not at all crucial. Meanwhile, USN's and RN's cruise missile capabilities were very important.
Air Superiority - Latest BVRAAM and SRAAM capabilites with high maneuvering for close in combat.
This role is beginning to be disputed by long range SAM's with CEC.
Air Interdiction - Precision ground attack weapons such as JDAM as well as BVRAAM capability.
Also doable with bombers for long range, and closer to shore with guided shells or sea launched tactical missiles. Did sea-launched ATACMS ever come to fruition?
Suppression of Enemy Air Defenses (SEAD) - Sophisticated electronic detection sensors and "fire and forget weapons" to target and destroy enemy air defenses including airborne fighters.
For some time yes, although I doubt this mission will be carried out in very near future more effectively via networked horde of UCAV's pinpointing targets for various munitions.
Close Air Support - Communications with forward air controllers and wide array of battlefield support weapons.
For Afghanistan-style conflicts UCAV's and bombers are more effective, for conflict against well-equipped power I don't see CAS having a chance.
Maritime Interdiction - Requirement for antiship missiles and datalink for passive approach and weapons launch.
On the other hand, with tremendous cost of aircraft, I would not think development of longer ranged ASM's should prove to be an insurmountable problem.
Also important for power projection are amphibious operations. There are aircraft carriers dedicated just for amphibious support.
I do agree with that role completely that amphibious role is relevant, but that may be filled up with various other ships such as LHA's.
All in all, I think CV's are terrific ships, but I doubt whether they're cost effective anymore. Carriers also have their limitations, notably short range of their strike aircraft if operating without extensive tanker support and quite low sortie rate.
Of course there's the issue that by operating a carrier all powers except the mighty USN have to tie their entire navy into defending it, thus making the carriers air wing (and perhaps a small number of cruise missiles) their navies only offensive weapon. The RN story where the fleet is drastically cut to buy carriers (for which I doubt full air wing will be ever bought) is an example of this phenomena.
For costs, one JSF is estimated to cost around 115 mil USD, while cost of a DDG-51 is around 1000 mil USD. Thus for a cost of a fighter wing of CV (F) alone one could buy, for example, four very capable surface combatants which are also extremely useful for stability operations, BMD etc.