1. MOST IMPORTANT, What are the key things you look for when looking at a used shotgun to make sure it is in good condition? I know how to check used pistols, but not shotguns, and I don't mind buying used if I stumble across a good deal.
It's the same kind of stuff, wear patterns, depth of wear, function of moving parts, function of the safety and slide release, barrel bulges. I was made fun of one time at a gun show when I shined a bore light down a shotgun barrel. "The rifling in there is just fine" he laughed. The heckler was ignorant. I could have informed him of what I was looking for, instead I left him stupid and moved on. On certain models certain things wear out. On the Mossy, check the condition of the safety button if it's plastic. On a Winchester check the thin spring that the bolt rides on. Check everything for tightness, but not to tight, know what I mean Vern? Avoid obvious awful gunsmithing. Missing pins, bulged barrels, etc. I think I may have a few more pointers on
this page.
2. I want a shotgun I can add tactical options to (foregrip, folding stock, accessory rail, etc) but still possibly use for bird hunting. Is this possible?
After building a few black plastic tactical whammo guns, I have come to the conclusion that a wood stock with a wood "cob" forend is my meat. Cut down the barrel, install a mag extension if need (I usually don't) and put of a sidesaddle for extra ammo. The one thing you will need for birds, depending on your state is a limiter in the magazine. Know your laws. Make your magazine where you can easily access it to reinstall the limiter. Truth be told, a second shotgun for HD is the better solution IMHO. Just get a good cheap one and keep it simple. You can arm yourself with a great HD shotty for $120 if you shop right.
3. Is it cheaper to buy a shotgun already put together with the options I want or to do it myself?
Almost as a rule, if you are going for the black plastic picatinny rail whiz bang stuff, it's cheaper to buy one already put together.
However.........if you think ahead, sell your extra stuff on ebay, and cut down a barrel yourself rather than buy a barrel, you can beat the gunshow tactical shotgun price (usually $250 or so) by a good margin. If you want cheap but effective, buy the ugliest gun in the pawn shop. If it has a fancy forend that overlaps the reciever, cut it down. If you can't tolerate the ugly finish, strip it off and stain the wood black with Rit dye.
4. Are the Remington 870 and Mossberg 500 the best options?
Another option is the Winchester 1200/1300 line. As HD shotguns, they all do very well. The Remington has a steel reciever vs aluminum, so it can take the constant pounding in the trunk of a police cruiser a little better. The Mossberg has it's controls in the right places. The Winchester has it's controls in almost the right places. I don't know what the folks at Remington were thinking!
Here's a page on the differences.
Expect to pay the going rate pluss a few bucks at a gunshow. If the dealer brought the shotgun with him, he expects to sell it, not cut the price. If you want the real deal, find it in the corner of a pawn shop gun rack under three months of dust. Also, Mossberg, Remington and Winchester all three made guns to be off branded by various stores. Western Field and JC Higgins are two common names. These are the same guns with a different stamp on them. They should be less expensive, but just as durable and have the same parts interchangebility.
Here's a cross reference.