Can the US Federal court system strike down a law passed by Congress and signed by the President?

Yes and no. Yes, the federal courts can review legislation the President signs, but someone who has standing to challenge the law must file a suit first (the Supreme Court can't just pluck a bill off the President's desk and start critiquing it). In order to have standing, an individual must be personally, directly and negatively affected by the law in a significant way and there must be an issue that can be remediated by a court. Not all legislation lends itself well to judicial review.