Hello, I'll try to help you 
remember this is a Brazilian kind of folk song. It takes place in the sertão, north Brasil. It's a very hard place to live because it has no water and it doesn't rain, the soil is dry, it is very difficult to survive and the people either go away or is poor.
1.Asa Branca. Is this a name of a bird?
I think so
2.Quando olhei a terra ardendo
Qual fogueira de São João-
a. Why do they use Gerúndio? I would use Particípio...
“Quando olhei a terra [b]ardida...”.
they use "ardendo" because when he looked at the land it was burning, in that precise moment
in Portugal we would say "quando vi a terra a arder"
b.“ ...a terra”: “to” land or “the” land? Or they've missed accent and it should be “à terra”- “to the land”?
you didn't miss any accent!
c.“Qual”= “like”? Is it possible to use it instead of “como”?
it's not quite "como"... a fogueira de são joão is supposed to be big, so he means the land was burning a lot, here "qual" is a way of emphasizing
3.Quando o verde dos teus olhos
Se espalhar na plantação
Eu te asseguro não chores não, viu
Que eu voltarei, viu
Meu coração
I can't translate this couplet at all... And what does it mean “viu”
here? (Generally “you saw”-I know this)
this is very hard

It doesn't mean to see, ver, it's a way of reassuring, something like "I'll be back, do you see, I mean it"
4.Please, don't laugh, but what about this situation with Rosinha? Our hero sends her to better place and fights with drought himself alone? Or she is from luckier family, which has their own water source, so he leaves and she stays? Or he goes to one place and she-to another and they'll meet again when good weather conditions return?
She stays there and he leaves to make some money, but his heart is where Rosinha is.
it's a way of living, men would leave and try to make money to help his family
5.“Até mesmo”means “even”. Is this specific brazilian or very modern expression? I've found it's meaning only in ONE on-line dictionary...
that's right "even the bird flew away"
6. Please tell smth about “sertão”.
at the beginning
7. And of course what is “fogueira de São João”?
It's a fire to celebrate São João, it is set in a religious day, but it's part of a great fest
8.Hoje longe muitas léguas
Numa triste solidão- This part is also beyond my understanding.
It means he is, today, very far far away and he is sadly alone
“Hoje longe”-what does it mean?
ok, here it would probably make it easier if it had a comma: Hoje, longe (a) muitas léguas (de casa, da Rosinha)
Muito obrigado.
P.S. Estou muito envergonhado pelo meu inglês.
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O teu inglês, assim como o português são optimos!
Espero ter ajudado, mas talvez haja alguém com mais conhecimentos que eu.
boa sorte!