A "Ceremony/Joining" Religious or Not is required to become married this is the legal part as you call it. There isn't always a need to obtain a marriage license, these are only required under certain circumstances.
The term / institution of marriage was adopted by the church and not invented/created by it, therefore they cannot enforce any of their doctrines other than within the church. While Catholics may have their own ideas of marriage they have no legal standing.
As for the child issue neither a husband or wife has any legal obligation to take on the responsibility of their new partners children unless they legally adopt or take on legal guardianship of said child. An example of this is when the biological parent dies their spouse does not automatically gain/take over legal responsibility for the child, who can be taken into care, handed to a member of the deceased's family such as a grandparent or even handed to the other biological parent.
I think you are correct about all of that, with the possible exception of the responsibility for the children. I have seen cases where the non-biological father was forced to pay child support even without adopting.