Chief Justice Roberts, dissenting, says (at p. 24):
The equal protection analysis might be different, in my view, if we were confronted with a more focused challenge to the denial of certain tangible benefits.
That indicates to me that there is probably a majority on the court that thinks same-sex couples should have the same adoption rights and other tangible benefits as opposite-sex couples, but that would have to be tested in court.
Regarding level-of-scrutiny, this opinion says nothing on the issue. But, given that Justice Kennedy follows a fundamental rights analysis, it could be argued that it is likely strict scrutiny, or at least a level of scrutiny higher than rational basis.
Vacco v. Quill, 521 U.S. 793 (1997), citing Romer v. Evans, 517 U. S. 620, 631 (1996) (emphasis added, internal punctuation removed):
If a legislative classification or distinction neither burdens a fundamental right nor targets a suspect class, we will uphold it so long as it bears a rational relation to some legitimate end.