If a man touches a woman with desire, this breaks his wudu, whether the woman is a non-mahram or otherwise. The kind of touching which breaks wudu is physical touching with no barrier in between.

If a man shakes hands with a (non-mahram) woman during the day in Ramadan, and he ejaculates, his fast is invalidated. He has to repent to Allah from this sin, refrain from eating and drinking for the rest of the day, and make up for that day’s fast later on.

It is not permissible for a man to touch the hands of a woman who is not one of his Mahrams. It is not permissible to shake hands even with a barrier in between such as shaking hands from beneath a garment and the like. The same ruling applies to shaking hands with old women.