The Auto-5, although a magnificent gun design from one of the best, John Moses Browning, it's not suited for continuous trap shooting.
Not being a trap shooter, I'm curious as to why you think this is so.
Prices seem to be all over the board. I'd like to know about options, model variations, which ones to avoid, etc.
These shotguns have been out of production since 1999, but were in production for most all of the century prior to that. I have shot examples made from 1936 all the way up to 1998 models, and found both highly satisfying. I prefer post-war units, and specifically those made after 1954-1955 due to the inclusion of the 'Speed feed' feature in units made after that date. If the gun can be loaded without depressing the bolt release, then it's a post-1955 gun.
The standard units are pretty beefy, and I've shot 200+ rounds in an afternoon thru the 1936 gun that I owned without discomfort. (I loved that 1936 - it was actually a pretty cool gun for range use.) The 'Light' versions (2 3/4" only; Light-12, Light-20, and Sweet-16) have small amounts of metal machined out of the receiver and barrel in non-critical spots, and are probably close to a half-pound lighter than an equivalent 'standard' Auto-5. I prefer them for upland work relative to the standard units, since they're a bit easier on the arms to carry about. The magnum units could take 3" shells and were built on the standard (not 'Light') receivers.
Production was largely in accomplished Belgium, with a break during WWII (during which they were made in the US) and with production being moved to Japan in 1976. The Japanese guns are certainly the equal of the Belgian guns in terms of fit and finish in general. When production was moved to Japan the barrels were redesigned with thicker walls and removable choke tubes; according to Browning, these barrels can be used with steel shot but the earlier Belgian barrels should not. The Japanese barrels are heavier than their Belgian counterparts due to the thicker walls; that is valued by some folk and less desirable for others.
It used to be that Belgian-made units cost more than Japanese units, not because they are mechanically better but in an 'Old World' snobby kinda way. These days, I see mechanically sound Belgian Light-12s with field-worn finishes selling for $300 while cleaner Japanese units still command $700+. It seems as if condition has finally trumped location of manufacture for Auto-5s.
In my neck of the woods, any of the Lights will sell for a premium ($100 or better) relative to the standard units.
Replacement parts are still readily available, due to the sheer number of guns produced and the fact that the design remained largely unchanged through its production run. Many parts require fitting (like stocks and such), making the guns seem more like a double gun in terms of maintenance than, say, a modern pump gun.
I know of none to avoid; these guns were't subject to constant re-tooling to address niche market segments. They are all designed to be working shotguns that will outlast any of us. The key questions to ask are relative to condition and intention - do you need a 3" magnum or a 2 3/4" standard or a 2 3/4" Light?.