The full manuals are not available for public distribution, but you can learn a lot about the Challenger's systems here:
https://www.smartcockpit.com/plane/BOMBARDIER/CHALLENGER 605.html
The page I shared with you is found in the flight controls document, for example.
I'm not sure about the tug freezing in place issue, but you might have more success with towing by powering the airplane up, turning on the HYD 3A pump, and cycling the parking brakes on and off in that state.
There’s a data output for brakes in your xplane settings - you can check to see if they’re actually on this way. Is it possible you didn’t remove the chocks?
I'm not exactly sure what's happening behind the scenes, but the checklist calls for the engine start to occur with the GEN switches off, then once the engines are running they are turned on. Were you able to resolve this?
There are three total pins in the nose gear area - one is for the nose gear itself, the other two are for the RAT extension and the gear door. Make sure you’ve removed all three then hit the switch to close the gear door!
Great! And just to close the loop on your question, the INFO message about ice on the wings will appear at the top of your screen in the sim, as long as your CL650 settings are configured to show INFO messages.