Yes, debate can help in various ways, but that really has a lot to do with how you approach debate in the first place. I did CX in high school and coached CX and LD for 7 years before coming to law school.
1. LSAT: being able to read complex passages on a diverse array of topics and quickly determine the structure of the reasoning. This, of course, assumes that you are tagging your own cards as a debater and listening carefully to the other side's evidence (and that you are getting practice with making arguments attacking the other side's evidence). I think the kind of reading you do in CX is probably more useful to prepare you for the reading sections of the LSAT.
2. LSAT: being able to think quickly under pressure. If your debate style is to pull out blocks and read the same topicality and politics DA every round, then you're not really training your brain in the right direction.
3. LSAT: as far as logical reasoning goes, I think debate may be helpful, but not in as concrete a way as 1 and 2 above.
4. Law school: being able to see various sides of an issue, hold both sides in your head at the same time and argue back and forth is very helpful on exams. But to get to this point in debate, you really need to focus on being able to extend arguments from earlier speeches, not just repeat them.
5. Law school: work ethic! Debate takes time and commitment. If you are able to put in long hours practicing and competing in an intense intellectual activity like debate, your first year of law school won't seem so bad.
6. Law school: not that it matters for your grades, but in law school you get called on in class, and some students really get nervous. My observation is that debaters really have more confidence and are not intimidated by a classroom of 120 students.
7. Law school: organizational skills. If you do your own research in debate, and know how to keep it organized, that will help a lot with legal writing and any other research assignments you get in law school. It will also help with organizing your outlines for class (but hey, you're only in high school, so don't worry about outlines!).
8. Life. Most debaters are pretty nice people, but there are some arrogant jerks in the activity. Debate helps you to deal with arrogant jerks in a competitive setting. As you may notice from this board, those skills are useful in the world of law.
I would say that overall CX can be a lot more useful than LD, but there is also a greater danger in CX that a debater will just focus on reading fast, capitalizing on technical errors of their opponents, and winning because their opponents just didn't understand the argumentation. If you really focus on the substantive aspects of CX (with emphasis on developing your own positions, cutting your own cards, tagging your own evidence, and really thinking during rounds), your competitive success may suffer in the short run, but in the long term you will be better off (in debate, and in law school, and in the real world).
And debate generally can make your undergrad years a lot easier, too. Good luck!