How do jet fighters get to war?
To answer the list in order:

Do they fly hop to hop?

An F-18 can go 3,000km unloaded (ordnance adds a lot of drag) and that's plenty from Australia. You can get to the Middle East in 4 hops, no air-to-air refuelling required. Although if you do an in-flight top-up you can add more than the regular range because you don't need to burn any fuel getting back up to cruise altitude.

Where would these stops be? Surely not at civilian airports?

Military aircraft often use civilian airports - they run on Jet-A just like everything else. Fighters were not an uncommon sight at the general-aviation terminal where I'm from. Loaded warplanes are a political issue, unloaded ones are a good day's business for the airport's fuel merchants. (note that fighters are almost never completely unloaded - the ammunition for the gun has a significant effect on the airplane's trim. But you can't see it.)

An external fuel tank can be added in place of weapons.

Are they actually shipped via regular military cargo ship?

Fixed-wing aircraft are typically not boxed for surface shipping. Helicopters are - no wings to remove and rotors can be folded up very easily.
Carrier-based aircraft will depart from land - the only way you get a Hornet off of a docked carrier is with a crane, so they typically launch everything when still offshore, and load the aircraft the same way on departure.

The USA has enough carriers that they would not deploy a carrier-based fleet like this. If they are adding / replacing aircraft there are probably enough already in Europe - send those where needed and replace them in the normal course of events.

Would they fly supersonic to reduce travel time?

No, they won't go supersonic to reduce the flight time. > Mach 1 consumes so much fuel you'd never get there. Fighters have supersonic capability so they can get there or run away as needed, but it's never used for very long. The Concorde was designed from the beginning for Mach 2 cruise, so it didn't use afterburners once at Mach 2. Subsonic flight suffered as a result, fighters would not accept the subsonic flight handling issues as that's rather important to them :-)

Bombers are often deployed directly from home bases. All modern strategic bombers have global range with in-flight refuelling. You don't leave North Dakota, blow up something in Longwayfromherestan and get home for dinner, but you don't land anywhere else either. Yes, it's many hours in the same seat but the military crews are used to it.