In my opinion.
Harebrained
Note the spelling. It's harebrained (as in, as foolish as a hare), not hair-brained. Just keep a particular long-eared cartoon rascal in mind to keep it straight.
As you write about Donald Trump's forays into presidential politics, feel free to use "hair-brained"; just recognize that you'll be making a homophonic pun. "Rug-rat" is a nice choice, too.
Hail vs. hale
More often than not, the word you want is hail.Hale means "being in sound health," and it often appears in the alliterative and redundant phrase "hale and hearty" — and it's "hale and hearty," not "hale and hardy."
Hale can also mean to "compel to go," most often in "haled to court." (While people complain about being hauled into court, haled is what will show up on official records.)
And, just to be complete, "I only regret that I have but one life to lose for my country" were reportedly the last words of Nathan Hale before being hanged by the British.
Everything else uses hail: You hail a cab. Hail covers your car in dimples. Barack Obama hails from
