we can answer this completely just by looking at the possible disjoint cycle types in A_{5}:

5-cycles (which can be written as a product of 4 transpositions: (a b c d e) = (a e)(a d)(a c)(a b))

3-cycles (which can be written as a product of 2 transpositions: (a b c) = (a c)(a b))

2 disjoint 2-cycles such as (a b)(c d)

the identity (which can be written as a product of 2 transpositions: e = (a b)(a b)).

which have orders 5,3,2 and 1, respectively.

the second part of your question can then be answered by asking how many 5-cycles, etc. there are in S_{5}, since for any cycle type in A_{5}, A_{5}contains every cycle of that type.