In Fullmetal Alchemist: Brotherhood, how come Alphonse can sometimes perform alchemy, and at other times it seems he can't? Also, early on the show, wasn't it said that he lost his ability when he lost his body? And if he can do alchemy, why isn't he a state alchemist too?
Alphonse can always do alchemy, throughout the whole show. The only times he cannot perform alchemy are the same times that Edward cannot;
When Homunculus uses his ability to seal off alchemy temporarily.
Now, in the beginning of the show, he does not perform alchemy often because he needs to draw transmutation circles to be able to. Drawing transmutation circles takes time, and is not often practical in fights or other scenarios that need quickness, especially if he does not know the proper circle (they can perform more basic transmutation simply, and both Al and Ed are geniuses, but they are still shown studying often). He is however, shown transmuting items throughout the show, both before and after
gaining the ability to transmute without a circle, like Ed does.
In fact, Alphonse has a particular ability (at least in the Anime) that no other alchemist is shown to have. He can transmute without physically touching the transmutation circle.
The first part of the question has been answered so I will concentrate on the State Alchemist portion. Alphonse wanted to be a State Alchemist along with his brother but Ed talked him out of it. The main reason was that being a State Alchemist is equivalent to joining the army. It means you have a commanding officer, you follow orders, and you go fight when and where you are told. As Ed told Al, "only one of us has to carry that burden".
Plus there is the issue of a health screening, and seeing as how Al is just a hollow suit of armor, their secret would be immediately blown.