This is largely a contract law issue, with little to do with intellectual property.
Let's take a closer look at Offer and Acceptance of a Contract. An Offer or Acceptance can be made:
- Orally
- In writing
- By conduct
First, you offer your client a written contract, with terms and conditions. Your client refused to accept. There is no contract.
Then, your client offers you another contract, orally:
We still want you to move forward with the project. We will give you $X
as deposit, and then you will begin work. After the project is
completed, we will deposit another $Y to you.
It is irrelevant whether $X and $Y are the same as in the written contract or not. This is a separate contract, and the original terms you proposed does not apply.
Did you accept this contract? From Felhouse v. Bindley (1862):
Acceptance must be communicated clearly and cannot be imposed due to
silence of one of the parties.
Since you accepted the deposit and began work, you have accepted the contract by conduct. This is a contract where there is no terms and conditions besides that one sentence mentioned by your client.
Furthermore, you intent to carry on this contract (you wish to complete the project and get the money). This contract has benefits to you for Consideration.
Would it have made a difference? Suppose that your client's response is:
We are sorry, our company has a weird policy about accepting contracts
in writing, 30 people have to approve it and it takes 3 months. Would
you take a verbal acceptance? We will give you $X as deposit and you
will begin work, after work is finished we will give you $Y, as stated
in the contract.
Then this would be an oral acceptance of the original (written) contract with terms and conditions. But that cannot happen, since the client explicitly refused your offer. An acceptance must be unconditional. Qualification of terms of offer means to destroy the original offer. Counter-offer means to reject the original offer.
Can the client use the deposit as evidence that there is a verbal contract?
Yes, certainly. But this contract is not the un-signed contract. Read this conversation, again, carefully:
We still want you to move forward with the project. We will give you $X
as deposit, and then you will begin work. After the project is
completed, we will deposit another $Y to you.
There is no mention that your proposed terms and conditions apply, is there?
Even if the client, after 2 weeks, come back to you and says:
Sorry for refusing your offer earlier. After due consideration, we think your company's proposal is best. We would like to move forward with you. It will be $X for the
deposit, and $Y after completion. All your proposed terms and
conditions apply. Deal?
This time your client is making an Offer. If you accept, it will be a new contract, not the same one you offered to the client earlier.
About the money. This is really a case by case situation. Depending on how much the entire project is completed and the nature of the project, results can be different.
A deposit is necessary to begin work. The deposit secures the client's intention to carry on the project. The final payment is make after work is finished satisfactorily.
Isn't the current situation exactly what a deposit is for?