At least the set-theoretical question can be answered: there the are uncountably the cardinality of the continuum many such series, as can be deduced from the results in this paper (not all of them attributed by the authors to themselves):
MR2293600 (2007k:30003) Reviewed Shmerkin, Pablo(FIN-JVS-MS); Solomyak, Boris(1-WA) Zeros of {−1,0,1} power series and connectedness loci for self-affine sets. (English summary) Experiment. Math. 15 (2006), no. 4, 499–511.

