この調味料（を）火（を）消したあと（に）加えてください。Using 〜あとに is a bit better in this case because you're doing something right after something. 〜あとで has the feeling of doing something later and the second action is a little bit further from the first action. i.e 学校のあとで買い物に行きます。can be like you're finishing school at 3pm but you're going shopping at 8pm, so there's a bit of a time lag, whereas 学校のあとに買い物に行きます。has the nuance of you going to shopping right after school.
授業の（あとで）、図書館**に行って宿題をします。placeへ is technically correct but if I want to be fussy I would say in this case に is a bit better.. へhas a nuance of going in that direction and I normally use it with like まっすぐへ行ったら。。右へまがったら。。
地下鉄を降りた（ら）忘れ物にきがつきました。[When/after I alighted the subway, I realised I had forgotten something.] I would think using たら here is the best fit.. if you use あとで, once again there's too much of a time lag and it doesn't sound quite right. あとに doesn't quite work here either because it more commonly used for a planned action..
(何も買わずに、すぐ帰りました。）is better.. [I went home without buying anything.]
試験の前**に（遊ばないで）勉強しなければなりません。I would use に here over は
Just my 5 cents.