1) I have not seen one, but you could proabably have one built. In fact I encourage it, support local craftsman.
2) Not that I'm aware of. The Scoop did a nice piece on Sauls place (The Rum Cigar and Coffee Shop in SP) which is probably the best known with the largest humidor, but I could find no basis for accredation as an authorized dealer with Habanos S.A....Havanas on Front St, however did. I was a bit surprised that Saul hung his hat almost exclusively on "Cubans" when there are many other cigars from the Caribbean that are noteworthy. My recollection is that he does stock a few local "Belizean" brands. I'll admit I havent tried them. Oh by the way the Airport Duty Free was not an accredited Habanos dealer either.
Wallen's Grocery and several of the Chinese owned "Supers" in Plancencia carry "Cohibas"(?) but my own trials with those sad crispy sticks left me cold regardless of authenticy. In fact I dont like Cohibas period; even the real ones. So I may not be the best judge of that brand from the stables of Habanos. The fact is youre going to see Cuban Cigars touted everywhichaway in everyewhichplace.
3) You take your chances. I can tell you categorically that if you have a cigar palate and you taste a true Cuban (Habanos S.A.) cigar vs a half assed fake and youre going to know it immediately.
There are plenty of Cuban seed cigars from Nicaragua, Dominica and elsewhere and many are quite good. I'd find a cigar you like and forget about whether its a real cuban or not. IMHO there are plenty of non-cuban cigars out there (but maybe not for sale in Belize) that beat the pants out of a true Cuban gorwn and rolled cigar.