Therefore, when a claimant, even a wicked one, has a claim against you, you must respond. In this case, the Shulchan Aruch has two opinions whether the Beth Din itself would consider the wife's claims, because she sinned by going to secular court. But while it is true that Ramo rules this way, and his opinion is usually decisive, in this case the Tumim who was a mighty genius and authority, completely disagrees with the Ramo, at least in some circumstances. So, if a Beth Din refuses to get involved, that is the Beth Din's business. But for you to deny the claimant an opportunity to deal with her claims is not smart, will result in a siruv, as it already has, and denies you the opportunity to turn the tables on her.
What you should have done is to respond to the Beth Din's demand that you have a Din Torah by agreeing to the judgment. But you can refuse to go to the Beth Din of the claimant. You can say that you will respond to the claimant in a different Beth Din. Or, you can respond that as long as she is in secular court, or as long as she refuses to relinquish all of her gains in secular court, you refuse to go to Beth Din. Now, the Beth Din may refuse you on some of these points. But you are not a plain refuser. You are a refuser with a reason. That makes a difference.
Now, if after all of this, the Beth Din comes after you, you respond to them that you want it in writing, that even though she won her claims against you in secular court, and only comes to Beth Din to finish you off by asking for a GET, that you must accept their demands. Say also, that even when the Beth Din reiterates its demand that you come to them for the Din Torah, you want another Beth Din. There are Beth Dins that are on the side of the Shulchan Aruch, even in such cases. At that point, the Beth Din must accept your decision if you have a Beth DIn that will back you. Rabbi Knoefler in Lakewood might be a good Beth Din for you. Once you have a Beth Din to fight for you, you are strong, and she is in trouble. The first Beth Din is also in trouble.
If your wife refuses the Beth Din that you chose, you enter the phase of zabla, meaning, each of you pick out one dayan. You will pick a Dayan and she will pick a dayan. Then begins the selection of the third Dayan, which is made by the two selected Dayanim. Hopefully, they will agree on a third dayan and they will settle all of the issues.
That is the basic answer for your questions. But let us finish off the other questions.