Law varies from state to state.

I am not sure where they live and there the property is. If they both in NY, here is what would happen:
第一，房子在丈夫的名下，如果以后他过世了

It depends on how much other asset the hu*****and has, the general rule is that the wife can get $50,000 and half of the rest, if there is no will. If there is a will, the wife can get at least 1/3 of the hu*****and's estate, regardless what the will says. Also it depends when the house is purchased, before marriage or after, and where the money comes from.

第二，如果房子是属于夫妻联合拥有（产权证上是两人的名字），

It is tenancy by the entirety and when one dies the other once has it by default. The kid does not have any claim.

Of couse, pre-nup or post-nup can change everything.