Hey there, welcome to the forum!

To answer your questions directly:

- does it work with DSM2/DSMX or only AFHDS?
- can I plug an additionnal servo to it or I will need another ESC?
- Is it the most recommended transmitter/receiver to go with the kit?
http://botbitz.com/product/hobbyking...tter-receiver/
- Do I need a soldering kit to link the drive motors and the ESC?
It will only with AFHDS but since that transmitter comes with a matching receiver, that shouldn't be an issue. Yes, if using the receiver supplied, then you'll be able to plug a servo straight into it. I suspect you will need a soldering iron to connect the motors to the drive ESCs. As for wherever the transmitter is recommended or not - the Devo 7e is normally favoured at the AW level (over in the UK at any rate) but that shouldn't stop you from using something else. My first AW doesn't use a Devo 7e, uses a completely different transmitter/receiver combination.

I have been told "ESCs are by far the hardest to find as you need to
find brushed ones with no brakes".

Does it correspond to the botbitz kit?
I am not sure I've ever heard that before but I don't see braking mention mentioned on the page for the ESC (http://botbitz.com/product/botbitz-1...ed-controller/) so I can only presume they aren't braked?

Botbitz provides motors with different voltages. But it seems that
most bots are using a 2s lipo battery which give 7.4v. Why would I need
the high voltage one then?
You wouldn't. The "low voltage" motors page on the site say the motors will support 8v so 7.4v LiPo's won't stress them at all.

For the batteries,
https://hobbyking.com/en_us/turnigy-...pack-5pcs.html
should be ok? With a lipo safe bag and a charger.
Looks good to me.

Regarding the transmitter, I found on one forum someone talking about
this one https://hobbyking.com/en_us/fs-i4x-4...io-mode-1.html
which is AFHDS too so maybe compatible with the botbitz kit. Is it
really compatible? Is 4 channels too little? Does it worth spending a
bit more on a better transmitter?
I couldn't say if it was compatible or not, presumably it is because of the matching frequency specifications but don't hold me to it. As for the number of channels, that is entirely down to wherever your design needs it. Your drive will need two channels (forwards+backwards and left+right) and for a weapon, I know some folks use the rudder channel to control it (I do on my Beetleweight).

If I go with
http://botbitz.com/product/hobbyking...tter-receiver/,
is it the V1 or V2? What is the mode 1 part of the description here?
It looks like there is a mode 2 also.
https://hobbyking.com/en_us/hobby-ki...v2-mode-1.html
Both transmitters linked appear to be V2s. The mode 1/mode 2 part defines on which side the non-sprung throttle channel is on. Typically over here, we use Mode 2 where the throttle stick is on the left, Mode 1 is where the Throttle is on the right.

Where can I buy the pieces to make the frame of the robot? It seems
that many people are using 3D printing now. But at the beginning I'd
prefer not to spend time on learning CAD and finding a printer.
I can't answer this one. I don't use a CAD printer myself for my AWs, I craft my frames with nothing more than a sheet of plastic and a Dremel. Fiddly but satisfying.

Everything is out of stock on hobbykind, is it normal and I just have
to monitor the products I'm interested by? Like
https://hobbyking.com/en_us/hobby-ki...usb-cable.html
for the transmitter
Yeah, you'll just need to keep an eye out for when it comes back into stock or find a alternate supplier.

What tooling should I buy except a screwdriver?
As hinted previously, I find a Dremel with a selection of drill attachments works well. If you don't fancy that, hand tools to cut plastic/metal would also work here, so a hacksaw would probably be in that list.

What good wiring guide should I read?
There are one or two "how to build a combat robot" threads around here that have wiriing diagrams with them, I'd go and find one of those.

For the servos for a flipper (or a hammer), I see there are different
possibilities like the number of degrees it can move. But I don't
understand how to read the other specification. For example, how to
know if it's powerful enough to move 200g?
Do you have an example of this specification? Also, unless the weight limit in Austrilia/Singapore is different for the Antweights, 200g is overweight (here, it is 150g).

Hope this helps.