For (1), the answer for finite (infinite) permutations is clearly yes, just by following the same proof for finite permutations. This of course gives a characterization of (finite) permutations. A permutation is finite if and only if it is the product of a finite number of cycles.
For non-finite permutations, we'll have to use an infinite product of cycles or transpositions to make sense of (1). So we'll need to define what we mean by this. I think a reasonable definition of an infinite product of cycles is that every element should appear in a finite number of cycles. This makes the product well-defined. With this definition the permutation $\sigma$ given by Michael can be written as $(12)(34)(56)(78)...$ which is well-defined.
This also answers part (2) in some sense since any permutation which can be written as a well-defined infinite product is the limit of its finite partial products.