"But the wind is weak today."
The subject is かぜ, so why didn't it get は instead of が？Would it be かぜは if 今日(は) was omitted?
because は is not the subject marker. the topic of the sentence is today (今日). the wind is the thing that is weak, so ig gets が
The topic of this sentence is today (and however it contrasts to whatever came before the でも) so it's marked with は
Yeah it would. "Today" is th subject. Personally, I read the sentence as: As for today, it is the wind that is weak.
I thought you can leave です (desu) out if the last word in a sentence is an adjective (-i/-nai)?
We can but only in informal speech. In polite speech, 丁寧語, the です cannot be omitted because it is the only thing that's elevating the sentence's politeness.
usually that one is used as a conjunction to attach the sentence to another one, similar to continuative form, but sometimes you can omit the sentence that you are attaching. And I think でも sounds more like a statement than using けど
It might rain today....
「 でも、今日は風が弱いです」but, the wind is weak today
do you wanna go fly a kite?
[you responding]「今日風が弱いですけど」but the wind is weak today (similar to how ね works and omitting the sentence that's trying to imply).
I would say that without context both are correct.
you can add it but is not needed, some temporal nouns have a particular property of behaving like adverbs and fall into the category of adverbial nouns as well. When used like this they can be placed at the start to modify the whole sentence by acting directly on the verb.
This is really similar to how you can use normal adverbs at the start of a sentence if there is no topic marker, for example「映画をよく見る」can be change into「よく映画を見る」and both are correct.
Sometimes however you will see a comma in between, I think in those cases is skipping the particle (which is also acceptable on written japanese but only for temporal nouns), and sometimes they use は to bring up the date as a general one, this is similar to how we use plural in english for example: "I'll watch movies on sundays" vs "I'll watch a movie on sunday".