For pork sausages, pork shoulder (Boston butt) has the right ratio of fat to lean, so you don’t have to add any additional fat to your sausage. For ground beef, chuck is the usual choice if you want something fattier (we’re looking at approximately 75-80% lean with chuck, depending on the trim). For hamburgers, chuck works great, but I particularly like to grind down (boneless) short ribs. They are BEEFY in flavor, and also have a good amount of fat on them. I’d guess similar levels to chuck. Brisket is also very good for burgers.
If you want something leaner, sirloin grinds up okay (try a top sirloin roast), as well any of the round roasts. The round roasts tend to be less beefy than the other cuts, so I don’t really recommend them, although top round is still reasonable. It’s eye of round that I don’t like.
For chicken, boneless, skinless thighs are best for sausages, patties, burgers, etc. Ground breast works okay, too, but has very little fat, so I only use it for stir-fry (although I usually just cleaver the breast into tiny pea-sized pieces rather than going through the trouble of grinding it.)
IMPORTANT TIP: Semi-freeze your meats before you grind them. I generally try to cut the meat into cubes (maybe 2"x2"x2") and place them in the freezer for about an hour or so, until it begins to get hard, but isn’t frozen through. It is also advisable to freeze all your metal grinding parts (the die, the blade, the screw or whatever it’s called, etc.) If your meat isn’t cold enough, what happens is the fat doesn’t grind properly and it begins to melt and gum up the grinding plate. The meat then is kind of squeezed like a paste through the holes and stuffed-up fat and you get a grind that is a pile of oily mush. You don’t want that. You want the fat to be distinct and the grind to have an “airy” quality to it.
For dog food, I just do 1/3 by weight of rice, meat, and ground or food processed raw vegetable like sweet potato or green beans.