I will both check whether the supply is actually connected to ground or not (yeah that didnt pop up to my mind at first), and also that the wire X is properly connected in any case, so that Im twice sure the current doesnt flow to the wrong side.
The "ground" on an Arduino is simply a common "reference connection" which may, or may not, be connected to earth ground. For example, if you have a USB-powered Arduino plugged into your laptop there is no earth ground. Even with the charger plugged-in there usually is no connection to the AC (earth) ground. But, it doesn't hurt anything to earth-ground the Arduino. And, nothing will burn-up if you disconnect a ground.
The "ground" in your car is simply a common-chassis connection, insulated from earth by the tires. 
In fact, if the Arduino isn't earth-grounded you could connect +5V to earth-ground and everything would be OK. But, that's generally a bad idea because if somehow the grounds are connected somewhere in the overall circuit, you'd have a short.
The power-line ground (earth ground) is for human-safety in case something "goes wrong". The chassis of your TV is (usually) earth grounded so something if goes-wrong inside the TV and the AC voltage somehow gets connected to the chassis, the power is shorted-out to ground, killing the voltage and blowing the circuit breaker before anybody gets hurt.
I assume breadboarding falls of in this case
I think the rule-of-thumb is about 1A max through a breadboard. One option is to solder (and heatshrink) directly the the MOSFET. But when you mount it (maybe to a heatsink) the tab is the drain connection so they can't all be mounted to the same heatsink.
Also, I wonder whether I should rather divide the project into eg. 3 parts, so that Ill rather have three 8A supplies than one so strong one
That's up to you, depending on cost & space, etc.
so I dont have 24A running in one cable.
With one supply or multiple supplies it's usually better if each MOSFET & solenoid has it's own separate wires (power and ground) back to the power supply. [u]Here is a wire gauge chart[/u] showing recommended maximum current.