  This paper provides upper and lower bounds on the optimal guessing moments of
a random variable taking values on a finite set when side information may be
available. These moments quantify the number of guesses required for correctly
identifying the unknown object and, similarly to Arikan's bounds, they are
expressed in terms of the Arimoto-R\'enyi conditional entropy. Although
Arikan's bounds are asymptotically tight, the improvement of the bounds in this
paper is significant in the non-asymptotic regime. Relationships between
moments of the optimal guessing function and the MAP error probability are also
established, characterizing the exact locus of their attainable values. The
bounds on optimal guessing moments serve to improve non-asymptotic bounds on
the cumulant generating function of the codeword lengths for fixed-to-variable
optimal lossless source coding without prefix constraints. Non-asymptotic
bounds on the reliability function of discrete memoryless sources are derived
as well. Relying on these techniques, lower bounds on the cumulant generating
function of the codeword lengths are derived, by means of the smooth R\'enyi
entropy, for source codes that allow decoding errors.
