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Yes, any time you send anything to the court you are supposed to send a copy to the other side. Normally you note that at the bottom of the motion in what is a "Certificate of Service" which just states that on a certain date you sent a copy of the document to the defendant and then you specify how you sent it (fax, certified mail, etc) and then you sign it.
Yes, the judge usually sets a hearing on the Motion to Dismiss and then each side argues their points. As soon as you recover from the surgery do any legal research you have to do and take three copies of the cases with you, one for the judge, one for you, and one for the other side.
It's not a huge issue, you can go ahead and send them one now and you should be okay although the judge may move the hearing date. Be sure and send everything by certified mail, return receipt requested in the future or some other method (a fax if they still use a fax machine) where you can prove they received the document.
If you sent them a copy before it was file stamped then it won't be an issue at all, just send them the file stamped copy and note on the cover letter that you are sending the file stamped copy of the one you sent originally that was not file stamped.