What is the ruling on a girl helping an elderly man cross the road, and is that considered haram (forbidden) requiring expiation?

The basic principle is that it is not permissible for a man to touch a non-mahram woman, except out of necessity, even if desire is absent. If the man is elderly and there is no fear of temptation, then it is permissible to shake her hand out of need. The expiation for unlawful touching is repentance, by ceasing the sin, regretting having committed it, and resolving not to return to it.