first all, don't run your grounds from battery to amps, have them all grounded to a single location...
2, if you're gonna run 1/0 , then run it all the way
3. upgrade the alt, that alternator might do for a little while but in the long run, it will just not be enough, your amps will kill itself for not getting anough juice
4. you could use that battery, but not recommended even if you build a battery box for it, get xs power, kinectik or whatever i don't know how much power you're gonna put out, you never listed the amps you're running