I have the 500mm f/4 L ii lens and it is a great lens. The autofocus capabilities are improved over the original 500mm L lens, but as previously stated by someone, unless you are doing a lot of hand held shooting, you won't see much difference over the original 500mm. If you are carrying it around a lot, it is lighter, but still not a light weight.
I would opt for the 1Dx in 95% of the cases. You will be getting a full frame camera with improved sensor and autofocus capability -- that will be noticed when using all your lenses. And if you are like me (and probably most shooters), you do most of your photography with other lenses. In a few months, rent the 500mm L ii and check it out. You may inherit a load of money from that rich (unknown) uncle and decide you also need the new lens.
If you have the latest firmware installed on the 5D Mark iii (version 1.2.1), you can get autofocus up to f\8. So, to use the 2X extender on a lens and still get autofocus, you must have a lens with a native f\4 capability. Since your lens is a f\5.6, only the 1.4X extender will retain the autofocus capability (1.4X loses one stop which will make a f\5.6 an f\8 with the 1.4X extender attached).
On a side note, you must install the extender on the lens prior to installing it on the camera (not install the extender on the camera then attach the lens) in order for the camera and the lens to properly communicate.
I have the 5D Mark iii and the 500mm f\4; the 2X extender used with this combination still provides good autofocus capability as an f\8 combination.
Wonderful that you are going to both London and Paris; Paris is my favorite city -- and I have traveled extensively in Europe, Australia, and the Americas. In fact I will be in Paris myself in April and plan to take some more photos.
I would advise anyone traveling in Europe to not use a backpack; a backpack and/or Nike shoes immediately identify you as an American tourist -- i.e. a target for theft.
I find that a couple pouches mounted on a Think Tank belt can carry all the lenses I need while touring around. This keeps them relatively safely stored yet easily accessible.
BEWARE: The lens they are selling is not the current lens, the EF 24-70mm f/2.8L II USM lens. They are selling the original version which sold for about $1400 new when it was the current model. So 42nd street is selling two year old models for a little more than they sold for when they were current.
When you speak of EF-S, EF, and L lenses, be aware that all L lenses are EF lenses. EF refers to the mount (attach interface to the camera). All EF-S and EF lenses will work on your 7D because it is a crop sensor camera. BUT only EF lenses will work on the 5D body (also 6D and 1D bodies). These are the 3 full frame cameras in the Canon line; all their other DSLR cameras are crop sensor cameras.
The L designation on an EF lens just means it is a better made lens (better glass, more solid build, more rugged); professional photographers will use mostly L glass. A side advantage is that L glass usually holds it value very well over several years if the lens is well maintained. I know people who have sold L glass for more than they paid for it after several years of use -- helped by the fact that Canon periodically raised their price for that lens.
Also be aware that even though your flashes will work on the 5D mkiii, the 5D has the capability to shoot using radio control also -- you do either have to buy an extra 600EX-RT flash or the ST-E3-RT which you insert in the hot shoe of the camera to activate the radio control signal. Only the 600EX-RT flashes have the radio control (in addition to optical) capability, but they are really great flash units.
First decide the brand (usually Canon or Nikon); the best way to choose is pick the brand that your or her friends are using. Both Canon and Nikon make excellent products. Then pick a camera and carry-around lens (not the kit lens!!!) that meets your budget. An excellent carry around lens in the Canon line is their 24mm - 105mm f/4; I'm sure Nikon has a similar choice.
By the way, the reason for choosing based on friends is that they can share equipment such as a lens or battery.