Only if 很 is stressed. Otherwise it pretty much acts only as a placeholder because adjectives under normal circumstances cannot be in place of the verb without a so-called adjective of degree (a word like “very, quite, completely” etc).
That's how you would say it in English, but there isn't really a way to only say "I'm healthy" in Chinese. "我健康" is not correct because it would translate to "I healthy" rather than "I'm healthy", so the Chinese requires the modifier and the English doesn't.
This question needs to be review Duolingo.
锻炼 means closer to "exercise training" like lifting weights and cardio. 运动 means closer to "exercise" or "work out" in any kind of sport. I really think you should be teaching 运动 instead if you want to go with "exercise"
The English doesn't sound very elegant to me. It sounds technically right, but I would replace the so with therefore. You could possibly also allow the translation: I am healthy, because I work out every day.