If my first pregnancy was high-risk for down's, will my second pregnancy be high risk for the condition?

Yes. The common factors that increase risk of downs include age greater than 40 years, and having a history of a previous down syndrome baby. Those risk factors will still be there for your next pregnancy. If your triple check test revealed high risk of down syndrome with a previous baby but that baby did not have down syndrome, then there is little increased risk for your next baby.
Yes. There are many new tests exclusive of amnio that can significantly lower the potential risk. Just went through this with my daughter. Discuss these new test with your OB doctor. Hope this helps. Good luck.