First things first: HMS Queen Elizabeth and USS George W Bush, plus their respective escorts, recently on exercise off the UK coast. A portent of things to come, perhaps. Note that a Type 45 destroyer has joined the QE group, I can see 1 Burke and 1 Tico, not sure if there are other US escorts outside of the pic.
does carrier size matter for these assignments? i mean, would a Supercarrier like we have put more planes up as part of the CAP than a small carrier would?
Yes
or would it be relative to the numbers of enemies you expect to encounter on a regular basis?
Also yes
(in this scifi setting the carriers run between tiny stuff with a 18-20 fighters and massive ones with 500+. even in space it seems likely that a carrier would keep a CAP up and have ready fighters. but their enemies tend to be as much if not more so fighter heavy than they are. i'm just trying to figure out how a naval aviation tradition would adapt to the setting's space warfare.)
Carriers are designed with these thoughts in mind; what is the purpose of the carrier and its aircraft, what are the activity requirements that purpose would entail and how to meet them. Of course in war, things don't always go according to plan. During the Falklands war, the RN deployed two carriers which operated initially 20, later 28 Harrier fighters between them. (20 in HMS Hermes, 8 in HMS Invincible; the former is a WW2-era ship not designed for jets while the latter is primarily a heli-carrier.)
The carriers were located 100nm east of the Falklands. A maximum of 3 CAPs of 2 Sea Harriers each made up the first line of defence for the British task force. Each fighter pair would fly out for (SWAG) 20 minutes to take up station near the Falklands, cruise for 30 minutes, be relieved by the next pair, then fly 20 minutes back. (The trips in and out were longer than distance dictated due to the need to disguise the location of the task force.) Flight deck operations were continuous; at some points all 18 CAP aircraft were in the air simultaneously. Pilots had only a handful of hours to rest between sorties; at peak tempo each fighter flew a combat sortie at least 4 times a day.
This is/was about the maximum combat tempo that can be expected, and not regularly sustainable - fighters just don't go out that many times a day normally. But that was all the British worked with. The full CAP was not always maintained, probably reduced to just 2 fighters at times when some Harriers were sent on ground-attack missions which they did during the initial period from 1st May until 18th-19th May when 12 more Harriers (some of them land-attack variants) joined the fleet. Though the British were really depending on their destroyers to stop aerial attacks, the Sea Harrier CAP was essential not just to look for Argentine fighters but also for surface ships, as their radars were powerful enough to spot ships 100nm away.
For the 500+ fighter carrier battles, you may need to think more in terms of WW2.
IF I recall correctly, at the Battle of Midway the US Navy had something like 12 fighters maintained on CAP and 24 at ready 10/15 (not sure) which launched only when IJN bombers were detected inbound. (The USN had the immeasurable advantage of radar.)
IIRC the IJN tried to maintain at least 24 CAP throughout the day. As a result, the initial USN bomber squadrons were slaughtered while the subsequent squadrons were untouched by IJN fighters who had by then run out of bullets; meanwhile the US CAP was more effective at intercepting the IJN fighters.
There's a balance to be maintained between recon, strike and CAP; each WW2 carrier had as a minimum something like 12 recon, 24 bomber/torpedo and 24 fighters ready for operations. The flight deck however could only "spot" or prepare for launch up to 40 aircraft at a time, and it took a lot of extra time to bring up aircraft from below to prepare for take-off, change the order of take-offs, or rearrange the deck for landing... see how crowded USS Saratoga is below: