If a the cop never files my ticket into the system do I still have to pay the ticket or is it just discarded?

That is very unusual. If the ticket never arrives at the Clerk of Courts, it basically doesn't exist -- regardless of what piece of paper the cop handed to you during the traffic stop. I would return to the clerk of courts and show them the copy of the ticket to make sure that they didn't misspell your name or that they didn't make some other computer error that could cause a nightmare down the road.

Two days isn't that long. The ticket will likely get filed sooner or later. I'm not sure what you mean about asking for an extension. The payment can't be due before the ticket is filed.

Unfortunately I have had to deal with this a lot. Officer's often take their time filing the ticket. Your ticket should have a court date on it and the officer should file it before that date. An attorney can file an appearance in the case using the ticket number and can start working on your case. Good luck.