it's not the power inverter, but yeh all inverters should be grounded most defiantely! The xbox doesn't have a earth, notice the plug? So yeh, you got to ground it properly like you would any other car stereo component, which by default the xbox isn't grounded properly (which i'm sure was a factor somehow when all those caught on fire a couple years back). i guess you wouldn't have to ground a toaster to the inverter, but the xbox is not a toaster (could be without a proper gound though).
Anyhow, i know you get it working right (don't forget the EQ). So what do you plan on using for the interface? A wireless controller? You can buy a usb 10key for like 8usd now days, that seems best if you are not using a touch screen. Also are you doing the 5.1 with a decoder of some sort? 5.1 in a car is damn near useless i know, but always something to play with :-).
BTW, what is the model/specs of the speakers you are using?
Also, i don't know what you mean by 12v screen...did i miss something?
Forgot, if you can, remove the hdd from the xbox itself to make it last alittle longer. All the bumps on the road will eventually make the heads loose or scratch the plates. So, really (follow me here), if you wanted to still run games, you could run them all off the hdd right. Well, then dvdrom is only good for what? Movies, data...etc basically non-xiso's. So, now you can use a PC dvdrom. This is good because basically you can chop up the xbox to half height (atleast half), and stick it somewhere with a long enough ribbon to the front console. Obviously being a ribbon that long (i think the spec says don't go longer than 18in), you might get something not working, but it's worth a shot. Mine for instance, i just totally removed the dvdrom all together. There is though ata133 ribbons tested for higher than 24", but i've never used one. Think about if you really need that dvdrom all the time or maybe just some of the time using it as usb in xboxlinux. Remember though, if possible use one of those shock packs (~8usd) around the hdd if you can.