Cross is a preposition, usually used with go, run, swim, etc.
Cross is a verb, which can usually be put directly in a sentence.
Crossing is crossing.
Prepositions like cross are usually used with go, run, swim, etc.
The difference between crossing and crossing is the direction.
Through is directly through the cross. Example:
I crossed the road.
I crossed the road (crossing the zebra crossing)
The car crossed the road.
The car crossed the road.