How do you know the other company did not sign the contract? Just because you do not have the signature does not mean that. Usually a party will say "return a signed contract to make it active" but that is not required.
Once a quarter is an enforceable action date in a court.
Did a lawyer draft this contract?
Interesting issue. As a general proposition, the other company must somehow manifest and communicate its assent to the contract before the contract will exist and become enforceable. Sometimes a contract specifies how a party must manifest and communicate its assent, and sometimes it doesn't.
That said, signing and returning the contract is one way in which the other party could've manifested and communicated its assent, but it isn't necessarily the only way. If your contract doesn't specify how the parties must manifest and communicate their assent, the other party could conceivably manifest and communicate its assent in other ways -- such as notifying you verbally, via e-mail, or via another type of writing -- or perhaps even by performing in accordance with the contract. If the other company hasn't manifested or communicated its assent in *any* way, then a contract hasn't been formed.
If the contract hasn't been formed, and if you're still interested in forming a contract with the other company, it looks like it might be a good idea to clarify some of those material terms.
Regardless of whether the contract was signed / delivered to you, have the two companies been performing under the terms of the contract? What is the specific concern you have?
Probably best to meet with a lawyer and bring a copy of the contract, all correspondence, invoices, statements, etc. He or she should be able to help you out.