I screw around a lot at the tech levels under discussion here, so IMO a carrier doctrine is workable but it's not particularly good.
The problem is that hangars (5 BP/HS) end up being one of the more cost-dense components comparatively, so carriers actually end up costing just as much as a missile cruiser of equal tonnage (~1,100 BP for 12,000 tons around INPE). This is a very bad position to be in, because a carrier needs fighters too, and those end up being twice or thrice as expensive per displacement ton. Additionally, reduced size launchers are only twice as large as box launchers and reload much faster - assuming eual magazine size, if 20 HS of x0.30 launchers are swapped out for a 20 HS hangar with fighters, each fighter would need to be 50% box launcher by volume just to break even, and you'd also need to stuff in sensor scouts. While carriers do have some extra free space since they can be slower and less-armoured than a missile cruisers, this doesn't really change the calculus significantly.
The end result is that a missile cruiser ends up costing less than a carrier and its parasites, while having almost as much throw weight and magazine depth, and being significantly faster and better-armoured. The only advantage that a carrier possesses is that it doesn't need to get into missile range to attack a target, and fighters are usually too small to detect effectively at typical missile ranges. This, incidentally, makes carriers really good for attritive warfare. I build carriers with smaller hangars and larger magazines, then fly multiple sorties using nigh-undetectable fighters to deplete as much of the hostile AMM defences as I can. Even if they end up doing almost no damage, they're still pretty good at softening up a target for eventual attack by a squadron of missile combatants.
Now, regarding your second question : there're two things that are essential for any good anti-fighter doctrine : robust defences against saturation strikes, and excellent long-range detection and targeting. The former more-or-less mandates AMMs, while the latter isn't as simple as 'biggest sensor possible'.
Regarding AMM defences, reduced-sized AMM launchers are a terrible idea and get outperformed by full-size launchers by a very larger margin. At this tech level, there are two options that make sense - long-range AMM fire with full-size launchers, and short-range AMM fire with box launchers. I tend to use both, typically with 12,000 ton mixed-missile cruisers and 4,000 ton missile destroyers .
Do note that it makes a lot of sense to invest in large resolution-1 sensors [~800 tons] if you're dealing with fighters, since you don't have to worry about the enemy cheesing detection by halving the size of their fighters or something. This has the added bonus of increasing your AMM range (I manage ~6-8 launches), which is great for dealing with saturation strikes. Additionally, scouts and sensor pickets are critical. I usually avoid having actives with a resolution exceeding 1 HS on any mainline combatants because of how noisy they are, and offload detection to fast 1,000 ton pickets that have the endurance and range to keep up with the fleet.
Since fighters are tiny, they don't usually have the space to have redundant MFCs to target ships and FACs - a 1,000 ton missile craft outfitted specifically for anti-fighter work does pretty well. Additionally, it's a good idea to keep scouts around that can shadow a fighter squadron back to its mothership - things tend to go really badly for a carrier fleet after this point.
To summarise, my general doctrine in this situation would be to use structure fleets around cruiser battlegroups, with 1-2 command cruisers with jump capability and large res-1 active sensors and oversized passive sensors, 4-6 mixed missile cruisers with large res-1 MFCs, deep magazines, and as many AMM and ASM/AFM launchers can be stuffed in, and 4-6 destroyers with AMM box launchers. This would be surrounded by an inner ring of six squadrons at the limit of the res-1 sensor's detection range, each containing an active sensor picket, a passive sensor scout, and a pair of anti-fighter missile attack craft. Beyond this, there would be an outer ring of passive sensor scouts, and independent free-ranging active/passive scouts used for reconnaissance. In general it's also a good idea to keep squadrons of box-launcher missile-frigates around to overwhelm the railgun cruisers, but you don't need many here since the enemy appears to lack robust AMM defences.
