I have an American Standard part #2603SW.011. The landlords installed it just before we purchased the house. I'm told he caulked behind, but there is no caulk I can see in the joints.

SOURCE: We have shower doors that have a clear caulking

Yes. Look for a silicone caulking with a mildewcide (some brands call it Bio Seal)

SOURCE: my tub surrounding is about a half a inch wide

You may want to try to find some thin ceramic tile beading and gluing it in place and caulking the remaining gap. Use a waterproof construction glue. This is most likely to be something you customize to fit. Often you can rent a ceramic wet saw and cut tiling to fit the need. Use proper safety gear, and be particularly careful if ripping thin pieces less you gain the nickname stubby.

SOURCE: i have an american standard jet tub, but i lost

You can adjust the jets by moving the center of the jet around, if the center does not move, it is not adjustable, some jets have a pressure adjustment by twisting the center of the jet left to right. No manual necessary for this.

SOURCE: the easiest way to rid our tub of black (mold??)

get a good tile cleaning spray before u use the spray first get lil buket put half water good mout of bleach in the water after the bleach add good mount ajax mix it round for 3min get scruber dip the scruber add good mount of the fluid scrub 4 good 6 min once dun add lil more fluid put were the black is let it set for 4 min then scrub once more it then it shud look lil clean then apply the tile cleaner spray the heck out of it were its black very good mout of tile cleaner spray let it set for good 3-4 min it sud look nice an clean. as 4 the calkin part its better to remove the black calkin parts wit screw driver genntly get new calkin add were the calkin was once u apply the calkin ur going to need damp rag once u put the calkin in the corner wet ur finger run it like if u were striking a match then the extra calkin after u run ur finger on the corner wip on the rag so fingers clean wet finger gain now the corners shud look smoooth an clean.

SOURCE: My newly tiled shower(app.2 yrs.

You will need to clean as much of the old caulking from joints that you can, After scraping old caulking out then wash with cleaner such as soft scrub and rinse thororghly. Let dry and wipe any residue with clean cloth.
The best all around tub and tile caulk and adhesive that I've found is called polyseamseal. It comes in almond which is pretty much the same as other caulks bone or beige. It applies easily and you can wipe off excess with fingertip as it is a latex base product.
What I do is apply as thin a bead as you need and wipe with damp fingertip, cleaning excess from fingers often. Continue wiping till you are satisfied with joint. VERY IMPORTANT: Keep completly dry for 24 hours. It will skin over in a couple of hours but it really needs the full drying time.
After it is dry there will be a light film around joint, use damp rag to wipe this film clean. It's not unusual for it to have a slight film on it more than once so don't worry about it.
I have been using this product for 30 years and if used right it has always made my customers happy. I hope this helps you. Thanks