An application for Cancellation of Removal is filed with an Immigration Court defensively, meaning that you have previously been placed into removal proceedings. At one of your Master Calendar Hearings before the Immigration Judge, you or your attorney will be ask what form of relief your will be seeking. At this point you will ask for cancellation of removal. Depending on your status, you will either file and EOIR 42A (for an LPR), or an EOIR 42B (for a non-LPR). Your Immigration Judge will normally tell you at this hearing when the application must be filed with the court.
You must file the application (and only the application) with the Texas Service Center, a copy with the Immigration Court, serve a copy to DHS Chief Counsel. The actual application with all supporting documents is filed in open court. Do not forget to file the necessary fees either.
You should really consult with a skilled immigration attorney before making any filing as any error could have serious implications and could result in a removal order.
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It depends on the work load of teh court. Normally, when you submit a Cancellation of Removal application, you would have to attend a hearing on your application. Most of the time, the judge makes his decision after the hearing is concluded. In some cases, the judge may ask for additional evidence before he makes his/her decision that may cause the additional time.
It is not the clerk or the judge that files the application. Rather, it is you or your attorney that is charged with that responsibility. If you are saying you filed at the window (instead of in the courtroom itself) the court would normally review the application at your next master calendar hearing and then determine whether or not it was filed appropriately. If it wasn't, you should be given an opportunity to cure defects. If it was, a hearing on the merits of the application should be set with you being required to make sure your biometrics are up to date.
Lastly, there is no specific period of time within which the court has to reject an application. As stated above, these decision are made at the hearings.