There is no way they outlaw birth control, they need a much more significant margin than they have at any level of government to even consider such an action
No they would likely make it a states right to choose (ironically enough). You got to look backward to get a clear concept of how conservatives like their activist judges to behave. When legalizing discrimination it's easier to make it a state rule and then fight it out there for three reasons. 1) Eventually all the conservative states will outlaw it which is what their base wants. 2) It would be a galvanizing issue they can use to keep winning in those states. 3) it's harder to stop discrimination on a state level than the federal level because they need to fight it state by state. Arguably the only way to stop it is for the federal level to tell them to stop.
It's only in regards Robbing people for big business and corporate interest that they like to make things national. For example breaking unions.
Also I got to just say this again, they didn't need to pick a candidate like Kav to push this through. They really need him to keep the Dems from forcing Trump to be subpoenaed. That is the only goal of this pick. Keep Trump (and maybe even his family) from having to testify in front of the house judiciary committee in a few months. All the rest of the evil was going to happen as soon as they got a conservative 5/4 judgement.
The only tiny silver lining is Kav is such a shit candidate and Gorsuch was such a dirty move. Packing the court is likely going to be treated as a legit option come 2020