Is it correct when I use " I'm not able to understand a word he's saying"?

Yes, OK but first is more common. 

    Is it correct when I use "The teacher told me he didn't manage to correct my essay because he didn't succeed in reading my writing. "?

"He didn't succeed is reading my writing". While maybe ok, and I know what you mean, we just don't talk like this as a matter of style/word choice. 

    Is it correct when I use " I'm not able to believe it. I got an 'A' in maths!"?

Not in this context, "I can't believe it!" in this context is more of an exclamation of surprise with a dose of doubt. 

I could say, "I'm just not able to believe that NK is serious about peace".

It seems to imply a more serious tone, like I really can't believe it.  

"I can't believe I got an A!" = 我不敢相信...!

"I am not able to believe NK peace offer" = "我不相信..."

Does this example make sense?

Yes, grammatically correct. But we normally say "I am unable to understand a word he's saying" or I don't understand

Number 2 is correct. The word "couldn't" refers to impossibility in the past.

Number 3 We normally say "I can't believe it! I got an A in Math"