Like the title says:

How do they ward off air and sea intrusions, spy ships, aircraft, etc?

Is there some distance (air and sea) that they declare?

Must be complicated, since The Group is in a constant state of movement as a whole (to include course changes), AND the repositioning of support ships within the group.

If it’s international waters, how does this work within international law?

Or, do the groups just operate unilaterally, and force incursions away by direct threat of action irrespective of sea and air laws?

I’m thinking the opening scene in Top Gun with Maverick and Goose for example.

Just wondered about this from time to time.

The exclusion area is centered on the battle group position. It moves with the battle group. How large that area is depends on the threat. Potentially hostile jets moving at mach+ speeds are going to have a much larger exclusion zone than a destroyer moving at 30 knots.

What sea and air laws do you think are being disrespected?

from the top gun example, so no real specific details are given up.
an air cap, or overwatch flight is always flying.
there is a ready flight on the cat ready to launch.
there are various ships spaced around the carrier to be missile suckers, one in charge of air defense, another in charge of underwater defense.

the whole carrier group as said above has an exclusion zone that moves with it.
depending on where it is also makes a difference in how that zone works.
when transiting through the straights of hormuse(sp) for instance, it is much smaller, but ships are generally at GQ for that transit in case someone gets froggy.

if there is a known threat area, then there will be more ready, but generally speaking, ships and planes that get too close get a warning on radio, ships sometimes even lights.
that air cap will go to intercept any planes that don't listen.
a DD or FF or CG will go to intercept any ship that doesn't listen.

for ships, I think I recall something like 5 miles. At least that is as far as a 5 inch can reach out and touch someone.
Though I know we manned 50 cals when boarding other ships so we were way closer.

As you mentioned aircraft have to be way further out, to tell them to go away as some missles have a couple hundred mile range.
but that is what missles and ciws is for if someone lets them get too close.

Interesting post, especially on this board, although I can't add anything to the discussion

FWIW, my late father was the XO of the Forrestal and the CO of the Oriskany "way back when", in addition to being a A-3 Skywarrior pilot prior to those assignments

20 nautical miles is the range of a 5" 54 mod 4 gun.

When I was transiting the straits of Hormuz we had 25mm cannons, started out as the loader, got moved up to gunner. Ain't collateral duties fun? Back then they were crew served, I understand today they are remote operated?

Of course my last visit was in 1990.

The effective range of AEGIS is 250 nautical miles. Too bad the Tico's are going away. Arleigh Burke is getting long in the tooth, I wouldn't be surprised if they start decomming them soon.
Harpoons were 80+ miles if I remeber correctly.
AEGIS motto was If It Flies, It Dies.
All
Enlisted
Gathered
Into
Servitude...

AEGIS is nothing more than a sensor package. Gotta have a weapon to reach out and touch someone. Harpoons are a surface to surface weapon, not going to see a surface target 80 miles, much less 250, away with whatever platform your AEGIS is on. It will have to use someone else's data to target anything over the horizon.