Most leases in Arizona specify 1) who is signing on the lease and 2) who is authorized to live in the property. The people who sign the lease are generally the only ones legally responsible for the rent payments. It is possible that your new landlord will either write in your girlfriend's name on the lease as an authorized occupant, or there may just be a clause that states a maximum number of people allowed to live in the property.
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Of course you can get an apartment with just your name on it. Just make sure the lease allows you to have a roommate, or, conversely, make sure the lease doesn't state that you are the only one allowed to live there. For a fairly nominal fee, I'll be happy to review your lease if you want to make sure you've got the flexibility you need.
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Certainly you can but the decision will be entirely upto the landlord. You should contact properties you are interested in renting and ask them about the policy. In my experience most properties are going to require a non-dependent to be listed on the lease as a contractually liable party. Also, while her credit may be bad you will likely still be able to rent but may have to put down a larger security deposit.