Yes, having a legal agreement in place would have been best but since you don't you have to work with what is in front of you.
I think you need to ask your attny the following questions:
The two of you had a verbal agreement to work through ALL issues (custody, asset allocation, support) through mediation. In NY a verbal agreement is binding. Now that she has breached the verbal agreement what are your options?
It is my understanding that when one spouse presents another spouse with a custody request or asset allocation request it must also include a petition for divorce or separation.
Since you and your W had a verbally binding agreement to pursue mediation, I would ask your attny about filing a petition to request immediate reimbursement to YOU from your W for your legal fees. And, be sure the reimbursement is not drawn on joint funds.
She has breached an agreement and therefore set the tone for this next phase. You both agreed mediation, she walked away without even letting you know, served you with papers and now you are forced to retain counsel when that wasn't the verbally binding agreement the two of you had. Think about that.