The stock speaker wire is fine - no need to run new wire to each door as that will be overkill. The only reason you would need to run new wire into the door is if you plan on installing components and not installing the crossovers inside the door.
Power wire - depends on your amp. 1/0 is not always best. If you are running a system that draws 300 amps of current, then yes you'll need 1/0. Check the fusing on your amplifier and check the owners manual as most will tell you what you need for that particular amp. The brand of wire isn't all that critical, just make sure that it is nicely insulated. Cheaper brand wire tends to lack a few strands of wire (not that critical for those not doing competition) and aren't as flexible as compared to more expensive brands. Other than that, it's fine for your typical install.