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.