"Is next week okay?"
I think both of those sentences could work in both of those contexts, but you're right in the commonality of those usages.
I'm still learning so take this with a huge grain of salt but I think genki is "okay" in the sense of "good health. You would use daijoubu rather than genki if,for example,someone fell over and you're asking if they're ok. Just like in English. From what I gathered from Japanese Ammo, another example is if you're doing some strenuous task and someone offers to help, you would use daijoubu to say "No I'm okay".
Also, in cases like these where "okay" means "okay with you" rather than asking if someone is physically okay, you can't use genki as it refers only to the "healthy" meaning of "okay".
Would appreciate if someone more experienced let me know if I'm mistaken.
You are right but missing some examples. For example if someone is being all jumpy and happy you can also use 元気 to describe them as lively, if you use it yourself it means you are fine as a response to "how are you?" like in a greeting. For 大丈夫 is more a word to use in a situation where you are worried about someone or about their opinion, if you use it yourself it means you are ok with the situation.
You can also search in google images 元気な人 and 大丈夫ですか and you will see some difference.
「元気ですか？」"how are you?" asking in a non worrying tone, probably as a greeting
「大丈夫ですか？」"are you ok?" asking about in a worrying tone
I think you can use it, but it's not the correct translation of the above sentence. 再来週はどうですか means "How about the week after next?" while 来週、大じょうぶですか？ means "Is next week okay?" but obviously both of those sentences are very similar in meaning, aside from the obvious difference of 再来週 and 来週.