For legal advice, ask an attorney. This is not legal advice.
In order to have a binding agreement it needs to be confirmed by the maker. (meaning it shows a date when the counter offer was received. In a Multiple Counter offer situation, the maker signs in Box 7. If it were a single counter offer, they would sign acceptance in line 6.
As stated below, your Realtor should be explaining this to you.
Escrow begins with one of the Realtors takes the signed purchase contract, with all applicable counters, with the deposit check, to an escrow officer.
To reiterate, you need to ask these questions to your Realtor or their broker.