Can I say,
(a) They cross the zebra crossing.
(b) They walked cross the zebra crossing.
© They cross the road using the zebra crossing.
(d) They cross the road by using / with the zebra crossing.
a) They cross the zebra crossing - grammatically correct, but within most contexts you would be more likely to need
They cross at the zebra crossing.
b) across, not cross
d) ‘with’ is not correct. You could use ‘on’.
One of the simplest, most natural ways of saying what you wish to say is:
They used the zebra crossing.
You don’t need to mention that they used it to cross a road, as it has no alternative use.