Here is the question i'm particularly interested in :
Let $f$ be a modular form, suppose we know the $a_p(f)$ for all but finitely many prime $p$. Is this enough to know the modular forms i.e. to know all the $a_n$ ? If it is true can you give a proof or reference ?
I'm also interested (but it's less important) in the following related question ?
Is there an algorithmic way to compute the $a_n$ ?
Is there a least amount of coefficients that you need to know to compute all the others ?
As posed, the answer is clearly "no": it's not enough to know all but finitely many prime coefficients. To see why, let $f$ be your favourite non-zero modular form (of some weight $k$ and level $N$); then $g(z) = f(2z)$ is a modular form of weight $k$ and level $2N$, and $a_p(g) = 0$ for every prime $p$ except 2. But $g$ is not zero.
If you know that $f$ is a newform, then knowing $a_p(f)$ for all but finitely many $p$ is enough to determine $f$; this is a deep and significant theorem (the "strong multiplicity one" theorem) and I don't think there's an easy way to make it algorithmic. (At least, if you don't know the weight and the level beforehand then you're definitely in trouble. If you know the weight and level, then there are algorithms which will allow you to compute all the newforms of that weight and level, and then you can just check to see which one has the $a_p$'s you expect.)