He is portrayed as a prophet and a messenger, but not the son of God. His birth without a father was explained as a divine miracle. He is, however, portrayed in a very positive light and was actually mentioned in the Qur'an more times than Muhammad himself.

As for the impact he had on the Islamic culture; I believe his stories and miracles are well-known and he is talked about more than most other prophets. And naming children after him is not something uncommon (He's name in Arabic is 'Isaa).

I hope that answered your question.