I think a cross-claim is used for bringing action against a named party on the same side of a lawsuit and counterclaim is for action against the plaintiff and/or someone else. I don’t know whether one can file it after the first answer. If this is something you want to consider, recommend talking to a licensed attorney.
Upon filing an answer, I believe you will be accepting that court has jurisdiction over you. If that happens to be a concern, recommend talking to an attorney before submitting anything.
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