Ok, first I’ll give my answers then look at the correct answers and update my post.
1. I think it means “very”.
2. 空気 I know from WaniKani – it’s “atmosphere”; 澄む I’ll learn at level 45, so I had to look it up at jisho.org; it’s “to clear”. 空気が澄んだ – “Atmosphere has cleared” – I’d translate it as “clear-skied” (not sure there is such an adjective in English, though), “bright”.
3. I’d translate this whole sentence as “Very bright March morning”.
4. I think it’s です/だ at the end of the sentence.
update
1. Got that one correctly.
2. I think I got it close enough.
3. As for using “bright” instead of “clear” – I think it’s close enough. But it seems, I confused the tenses. As the end of the phrase was omitted, I assumed it’s in present tense, yet the official translation is in the past tense. I’m going to look this thread more carefully and if I can’t find anything – ask a question.
4. Got that one correctly.
second update
It seems I’m not the only one who thinks that this sentence could be translated in present tense as well as in past tense. I’ll need to ask this question.

• This reply was modified 8 years, 1 month ago by  trunklayer.
• This reply was modified 8 years, 1 month ago by  trunklayer.
• This reply was modified 8 years, 1 month ago by  trunklayer.
• This reply was modified 8 years, 1 month ago by  trunklayer.
• This reply was modified 8 years, 1 month ago by  trunklayer.