1) The literal translation would be something like: I mean, even if I may have said some things incorrectly, I'm assuming that I'm more or less succeeding in giving an idea about the truth of it.
(The sentence is talking about the truth of something although I'm not sure of the context so I don't know. You probably know it though so that's fine)
So let's break down 'başarıyorumdur'.
başarıyorum = I am succeeding.
başarıyorumDUR = '-dur' can go with 'herhalde' which means perhaps (it doesn't have to though), and it gives the meaning that I'm probably doing this, or I'm probably succeeding.
'Cevabı anlatmayı başarıyorumdur' = I'm probably succeeding in explaining the answer.
'Cevabı anlatmayı başarıyorumdur herhalde' = The same thing, but it gives more emphasis on the probably, and makes it seem like self-doubt at the last minute.
And doğrulu yanlışlı literally means 'with correctness and with mistakes'. It doesn't make sense in English, but the translation in context is in my translation of the sentence you gave.
2) First of all, you can't find the meaning from just looking at 'kaldığımdan olacak'. You have to look at it as 'ZORUNDA kaldığımdan olacak' because it derives from 'zorunda kalmak' which means to have to do something.
I'll break the sentence down:
Büyümek zorunda kalmak = To have to grow up
Büyümek zorunda kaldığımdan olacak. = It's probably because I have to grow up.
'-ndan olacak' can, again, go with 'herhalde', or even 'büyük ihtimalle' (most probably).It basically gives the meaning of 'it's probably from/because of this' or 'it's probably from/because of that'
And as with the other one, this sentence is also contextual. It suggest that the reason behind something mentioned previously is probably because he has to grow up.