We use the Lavievert board and case - bought it on Amazon. The trays are nice but too small. We use cafeteria trays with fabric attached with double sided carpet tape. We have it about 2 years, starting to show signs of wear. There may be better products, hopefully others will comment.
Lavievert 1500 Pieces Jigsaw Puzzle Board, Portable Puzzle Case with 6 Sorting Trays, Puzzle Caddy Puzzle Storage Saver with Non-Slip Surface for Adults and Kids
You can get good puzzles at many stores, Amazon, Target, Barns & Noble etc. . Some of our favorite brands include Ravensburger, Springbok, Buffalo, White Mountain, Vermont Christmas Company, SunsOut. There are other brands, hopefully others will comment.
We also use a cheap, "Lifetime" brand 48 inch fold in half round portable table, it is not very sturdy but does hold everything we need.
Good Luck, looking forward to seeing some pictures of the completed puzzles.
Good questions, I never thought of having rules.
On edit: After much thought, another question - if you buy a used puzzle and the previous owner separated all the edge pieces into a plastic bag, do you mix them back in to sort again, do you even sort at all? It is interesting to learn how others do it.