Sort a string, sort of
Pyth, 5 bytes

s.T.g


Test suite

Very straightforward: Group and sort, transpose, concatenate.

s.T.g
s.T.gkQ    Implicit variables
   .gkQ    Group the input input lists of elements whose values match when the
           identity function is applied, sorted by the output value.
 .T        Transpose, skipping empty values. This puts all first characters into
           a list, then all second, etc.
s          Concatenate.