As you are now aware, you can have a driving record started for you off of your social security number (SSN) if you are caught driving and are cited for traffic violations (as well as driving without a license) if you do not yet have a driver's license to post the violations to.
Kansas and Nebraska are both members of the Drivers License Compact (DLC) and Non-Resident Violators Compact (NRVC). The DLC requires member states to report traffic tickets back to the state you are licensed in, in this case the state which you would be licensed in if you applied and received a driver's license.
The NRVC requires member states to suspend the driver's license of motorists that receive traffic tickets while out of state and fail to pay them or otherwise comply with the citation. So it would appear that it is under the NRVC that KS courts informed the NE Department of Motor Vehicles of your failure to pay for the speeding ticket and any other citation you received while in Kansas.
So yes, even without a driver's license you can be issued tickets, have a driving record and get your driving privileges suspended. Suspending your license in this instance likely means not allow you to apply or be issued a driver's license until you take care of the outstanding tickets.
To find out for certain what you need to do in both Kansas and Nebraska to take care of this issue you will need to contact the court listed on your KS citation and the NE DMV. In all likelihood you will have to pay for the tickets you received in KS as well as the reinstatement fee. The reinstatement fee may not be reinstating a license in this instance but instead taking the suspension or hold off your driving record so that you can apply for a driver's license now or at a later date.
If you are able to at some point obtain a valid license and need to acquire auto insurance you can click here for an affordable car insurance quote.