Drywall right up to support box?

I have the exel chimney system with the square support box which extends down into the living room 3 inches. The surrounding ceiling material is drywall. The trim plate covers the places where the drywall meets the support box. Is there any concern with having the drywall touching the support box with the paper backing on the drywall?? Do folks run the drywall up tight to the support box and caulk it, or leave a gap and caulk it, or leave a gap and not caulk it?? And if caulk should be used, is there a standard high heat caulk that should be used in this situation? And do you caulk the trim plate also? (The pipe is the 6" chimney pipe and the stove pipe is double wall.)
Any advice appreciated.
Thanks

Thanks to begreen and fossil for the responses--always nice to get some reassurances when you're thinking of safety issues, and scenarios and options are just bouncing around in your own head,