"Is next week okay?"
Why is there no particle は used? Next week is the subject and topic here, no?
は doesn't really influence the level of casual/ formal. It's just unnecessary in this case and can be omitted.
俺は山崎だぜ！Is definitely casual and uses は.
お疲れさまでした！Is definitely formal and doesn't use は because it's unnecessary.
It seems like the comma is used in place of the particle, to imply a more casual dialogue. I might be wrong though.
Would ”来週はいいですか？” not be a better translation for this? I thought "大丈夫" was more for enquiring about wellbeing?
大丈夫 is a very useful word that can be used for a number of different meanings. It basically means Ok.
That would mean "is next week good?" rather than "is next week okay?". The word choice indicates different contexts, for example looking for a suitable date to visit a friend versus asking for help to fix your fence (hoping the date won't be too inconvenient).
i'm not an japanese expert, but i've seen a lot of sentences using daijoubu, just like that one.
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