Your post is actually two questions.
As regards the first question, about asking permission, I would say they are interchangeable and refer you to @Yuuichi Tam's answer.
As far as your question applies to using either 「大丈夫です」 or 「いいです」 as a reply to a question (even one involving said words), it is often better to use less ambiguous terms. Both 「大丈夫です」 and 「いいです」can have dismissive or even negative tones, similar to 「結構です」. To signify the affirmative by affixing a positive expression before these terms, such as using 「はい」 or 「うん」 would be better.
Even answering a question about your health with either expression can sound a bit abrupt, as if you are trying to steer the conversation away from that topic.
If you incurred a minor injury and someone asked you 「大丈夫ですか」 it would be appropriate to use 「うん、大丈夫です」 as you are indeed trying to deflect attention.
However, answering the question 「体の具合は大丈夫ですか？」 with 「うん、大丈夫です。」 still sounds a little short to me. I would recommend using something along the lines of 「お陰様｛かげさま｝で元気です。」 or just 「元気です。」
While it was not specifically addressed in your question, answering an offer question such as 'Is it alright if I bring you some water?' with either expression might sound confusing for the other party and likely sound negative. You might get a befuddled look.
「水を持ってきてあげましょうか？」
「大丈夫です」or 「いいです」 - 'It's alright.' or 'I'm fine.'
In these cases, saying 「はい、お願いします。」 or a simple affirmative should be much clearer.
