I don't know the full answer to your question, but I'll tell you what we do with the caveat that there may be local differeces.
Do parents still make the sign of the cross over their children, going from left to right?
Yes, all the time.
NOT "really' blessing as a hierurgy, but it is a form of prayer. In our tradition it is not only allowed, but is sort of mandatory, especially when the children are small. We pray "the baby", and we make the sign of the cross. We do NOT say "be blessed" or "God, bless your servant ___". This is only reserved for priests. At a minimum we say "In the name of.." and "God, guard him/her". We usually say either the appropriate prayer (for instance the prayer of parents for their children from our lay prayer book "Lord, our God, Who with wisdom didst create man out of dust..." or the little half prayer half nursery rhyme "Little angel of mine, whom God gave to me.." or simply the Lord's prayer). What is also practiced and severely frowned upon by the Church is a special "disenchantment spell" against the evil eye ("Go away Evil Eye, to a land without greenery...").