Paranoia is a game in which everyone is or appears to be secretly working against you, and you are aware of this. (That is why it is called "Paranoia"). Secret societies and mutant powers are the main way in which the game produces this.
Mutant powers give you extremely useful abilities, but they're also things you could be declared a traitor for if you get caught using them. Secret societies offer convenient advantages for members and / or threaten to kill you if you don't obey orders, but the missions they send you are invariably things that could kill you if they catch you. Both are things you want to keep secret at all costs, but that other players could easily discover if you're not careful.
Of course, this means that you have a strong motivation to discover the secrets of the other players, so that you can use them against them. And they know you will come, and you know they know you will come, and that means that the conflict and intrigue are very likely to escalate until one or more of you are killed.
Tl; dr: Be alert. Trust nobody. Keep your laser handy.
Anyway, to answer your question: There is no way to choose not to have a secret society or mutant power, because the game doesn't work as well without them. Except for the intervention of GM, there is no way of "choosing" for the mess of intrigue that makes the game fun. (Multiple editions of Paranoia tell GM to tell players that not having a secret society or mutant power is a possibility. This statement is an important part of the game and should not be omitted.)
The consequences of not fulfilling the missions of your secret society depend on your superiors in that secret society (that is, even the GM). Sometimes this can mean punishment of some kind; other times it may mean that you do not get the contraband equipment or useful information that would otherwise have been your reward. There may even be no consequences at all. Or at least, there are no consequences to say you about…
The consequences of not using your mutant powers are that you don't get the benefit of using those powers. Of course, you also don't risk getting caught using those powers, and don't risk those powers going terribly wrong somehow.
Oh, and you mentioned that it is really difficult to figure out what information you are allowed to have and what information you are not. It is true. Accidentally revealing that you know too much is another source of in-game hilarity. Therefore, a healthy attitude for a citizen of the Alpha Complex is "I don't know anything," sometimes even when someone just asks for their name. And, on that note …
SECRET SOCIETIES DO NOT EXIST, CITIZENS, AND THE CAREFUL GENETIC TESTING OF THE ALPHA COMPLEX MAKES MUTATION A PRACTICAL IMPOSSIBILITY. IT IS THEREFORE IMPOSSIBLE THAT YOU ARE A MUTANT OR MEMBER OF A SECRET SOCIETY. THE IMPOSSIBLE BEHAVIOR IS TREASON. IF YOU BELIEVE SOMEONE AROUND HAS DONE THE IMPOSSIBLE, PLEASE INFORM HPD & MC FOR MANDATORY MENTAL RECALIBRATION.