Are marriages not concluded in Church valid in the eyes of the Vatican? Will the resulting children be baptised?

Answer:

If only one of the two partners is Catholic, the marriage has to be solemnised in a Catholic church, otherwise it is not valid (cf. Can. 1117).

If the marriage is not valid, the children cannot at first be baptised. There will be a postponement of baptism (cf. can 868 § 1 n. 2), until it has been ascertained that someone (relatives, godparents etc) can take on the Catholic education of the child.