Can I be charged with a DUI when the cop never seen me in the car?

If charged with dui someone must be able to testify that you were driving and that you were under the influence. If no one from the school saw you...

Consensual sex among two consenting adults is not a crime in Missouri. If one party is a minor they cannot legally consent.

Bought a used car from a dealer and was not given a title, now it is lost, what can we do?

I would call the Missouri Department of Revenue and see if they are able to issue a duplicate title.

What are my options for a speeding ticket (first one)?

Yes, contact a local traffic lawyer and see what options are available to you for keeping this off your record. Hope that helps.