I appreciate the fact that you are doing more than the average pro se litigant and are actually getting involved in the discovery process. However, I would suggest that these issues will ultimately provide you little relief.
As to the contract, while it would help their case to have a signed contract, it is not required. They can prove a contractual relationship by way of your testimony or even by way of the fact that you clearly used the credit card.
Similarly, I don't see the unjust enrichment argument getting you very far. The definition of unjust enrichment is a benefit for which the one enriched has not paid or worked and morally and ethically should not keep. Unfortunately, your situation does not meet either part of this definition. First, they did pay for the benefit. Second, there is nothing morally or ethically improper here.
I regret that my answer is unfavorable, but please understand that it would be unfair to you (and unprofessional of me) to provide you with anything less than a truthful response.
Please let me know if anything requires clarification.
Also, several customers have asked how they may direct a question to me in particular. If you specifically want me to assist you in your legal matter, just put "FOR JOSEPH" in the subject line and I will gladly pick up the question as soon as I am on-line.