Does it matter if someone else's name is listed on the billing address if the payment comes from my account for business purchases?

That issue would not be apparent on your return. You can explain any discrepancy in the unlikely occurrence of an audit. 

If you are audited, the burden of proof is on you to prove your business expense deductions to the satisfaction of the examiner.  While it is certainly easier to provide this proof if you have a business-only bank account, a business-only credit card, and so on, it is certainly possible to provide the proof when you use accounts for more than one purpose.  You just have to be aware of the need to provide that proof if audited, and keeping your business records in good order for at least three years after you file your tax return.