What I've read does not indicate a dismissal with prejudice, but simply a withdraw of a petition and a dismissal of two cases (and explicitly says "without" prejudice in three places.) Furthermore, the language in the order regarding your obtaining substance abuse treatment could be used in further proceedings to show that you were at fault - and could persuade a judge to grant a DVTRO if there is an allegation that you relapsed. Perhaps the dismissal was kicked back because the lawyer added language similar to this in the stipulated dismissal. You might ask for a copy of that dismissal to see if she added any language.
The main problem with this is that she is dismissing two cases with one stipulation. I assume that your DVTRO case and your dissolution case have two separate cause numbers (I have never seen it any other way) in which case you need one dismissal for the DVTRO case and one for the dissolution case. If you file this in the dissolution case, the DVTRO under a separate case number XXXXX NOT be dismissed.
You should assumed nothing has been dismissed until you get a conformed copy of the dismissal order with the judge's signature (or stamp signature). In my experience, non DVTRO cases (including dissolution cases) can be dismissed by the clerk by agreement of all the parties. You would submit the stipulation and the court would process it when s/he could. It can take months. DVTRO cases are different and a hearing needs to be set. You MUST follow the procedures in the link above to dismiss the DVTRO.
It really does not matter that the stipulation calls your husband "father" and you "wife" as long as it is clear who is who. Also, the first of the date filing should be corrected, although it would probably not cause a problem if nobody caught the error.
Glad to know the information I've provided has been helpful! Let me know if you have further questions.