Camping along the nullarbor you'll have no dramas camping with dogs provided your happy to camp in the "rest areas" - you might have a problem at servos though - some of the servos dont like dogs! So when taking you dog make sure you give him or her a "pit stop" before or after reaching a servo.
You MAY have trouble down in Esperance finding somewhere to camp with dogs - its mostly national park camping.
You'll have a ball the trip across west is awesome I'd drop anything to go back across.
Kel and I did the trip across in 4 days straight to perth but we could easily have spent weeks getting across - there is just SOOOOOOO much to see and do out there.
Also get familiar with the two raised fingers wave - its a nullarbor tradition. Drive with your lights on at ALL times. Dont speed (too much) the fuzz are out there and we saw a few people get busted. Grab a UHF - even if its only hand held (though a properly fitted unit with external antenna is better) they come in handy for communicating with trucks and other drivers. The truckies are great and will let you know when it is and isnt safe to overtake. They know how big they are and they can see alot further in front than you can so they can give the best information. They'll even give you a heads up for cameras and radars too
Exploroz generally has pretty upto date fuel price info and I suggest you work out what sort of range you get to your tank(s) if you dont already know - theres generally about 200k's between towns though there are a few longer stretches.
If running a diesel vehicle use an additive like flashlube - it'll increase your MPG but more importantly it'll help if you get a dose of Dagnabbit's fuel - theres plenty of it out there.
Sorry I cant help too much re: the dog situation but i'm sure you'll find plenty of places to stay - a lot of grey nomads travel with their puppies.
Enjoy the trip - its a GREAT experiance.
PS Australia is HUGE!!!