Are we -the buyers- supposed to sign the contract also?
Usually (and I stated it such in 6.) buyers 'sign' with their money, i.e. implicitly accept the contract by transferring the funds.
But it also does not hurt to explicitly do. Only thing from contractual side is, the contract becomes active (i.e. buyers can transfer their funds) as soon as seller and escrows signed - while buyers signatures are not relevant for that state change.