Does the Supreme Court review laws after the President signs it?

Neither. It doesn't review law at all. Courts don't review anything.

Like any court, it can only judge cases, so it is no position to say anything UNLESS an actual case involving this law comes before the court. THEN it can judge the case. And THEN it might decide that the law itself is unconstitutional and therefore, in its judgement, it rules against this law in order to achieve justice. That judgement is new law in accordance with the doctrine of judicial precedent and renders the offending law invalid.

Take Obergefell v Hodges, the gay marriage case. Laws against gay marriage had been around for years. Congress passed the Defense of Marriage Act to make federal law against it. And those laws stood because the Supreme Court had never had to decide the issue, until the Obergefell case actually came to it on appeal. Only then could the Supreme Court make a ruling that banning gay marriage goes against the guarantee of equality for all citizens in the 14th Amendment, and therefore Obergefell (and the other plaintiffs) should be allowed to marry the people they want.

And that is why unconstitutional laws can stick around for years. The Supreme Court can't do a thing until a real live problem about that law comes up that it needs to make a judgement on.

NO. The Supreme Court ONLY reviews laws is someone affected by the law sues claiming the law is unconstitutional.

They never specifically review it in that sense. IF the case is brought before them then they will review it. However until that law is passed and someone takes it to Court they aren't going to get involved.

No. The Supreme Court only hears cases on appeal that have been decided at the Appelate Court level. The loser has to petition the Supreme Court who then decides on Constiutional grounds.

the supreme court reviews rulings made by lower courts.

They only review it if a case involving it comes before them. They are not part of the legislative process.

They only review laws that have been challenged by a lawsuit.

No they don't but bear in mind that a huge number of politicians are attorneys and they know the laws pretty well.