When one looks at the output of
gp.stabilizer
if gp is a well defined permutation group, than one finds a record containing
many "transversals". How should these transversals be understood? E.g. they con
tain
often many "extra comma's".

The transversal is realy what is known as a Schreier vector.
The extra commas come from the fact that S.transversal[x] is only
bound if x lies in S.orbit. If so, then

while x <> S.orbit[1] do
        x/S.transversal[x];
od;

will always terminate (and reasonably quickly).

To put it another way, from S.transversal you can compute,
efficiently an element s in S such that S.orbit[1]^s = x, for any x
in S.orbit.

Is there a procedure in GAP to solve the "word"-problem for permutation groups?
E.g. something like "wordsolve(element, group, group.generators)" ?

Not as far as I know, though it might be possible (I'm guessing) to
do something by defining a homomorphism from a free group to your
permutation group and the looking for a pre-image representative of
the element in question.

Steve