In IL you do not need any paperwork to transport a firearm IF you can legally posses the firearm in your residence state. However, it must be fully enclosed in a case and the mags are not required to be separate from the handgun but the gun must be unloaded. Recent court decisions have indicated that the consol and the glove compartment can be considered a case. You can also keep a firearm in your hotel room if there is no local ordinance prohibiting it (Cook County). The critical question here is how was the firearm carried into the room and transported in a vehicle. Those actions must be done lawfully, no CC or OC is allowed in the state, with a couple of exceptions.
Hope this helps.