It seems to me that the hyphens in "first-order conditions" and "second-order conditions" are correct, but I'm editing an article that also mentions "higher-order conditions" and "lower-order conditions." Should these phrases be hyphenated too? For some reason, it doesn't feel quite right.
Yes, you have to maintain the parallel structure and keep it consistent across the entire piece.
In short, there are different rules to this, and exceptions as well, but in this case, where the terms higher and lower are directly connotative of your actual words, i.e. first, second, and tenth, you should maintain consistency.