Doesn't "att" fit between börjar and bli?
Yes, you can write it with att in there and that's just as perfectly fine a sentence. However, it is often omitted, especially in less formal contexts.
What would be the difference between saying "Pojken börjar bli trött" and "Pojken blir trött"?
Kan man säger:
Pojken börjar få trött?
No, that doesn't work. It sounds like if you'd use 'receive' in English.
In a multiple choice I choose "Pojken börjar att bli trött" and Duo said it is wrong. According to @Emil with "att" is a fine sentence. Should I report it?
Why is "Pojken börjar vara trött" not accepted?
I put "Pojken börjar tröttna." Is that incorrect?
Tröttna does not mean to get tired as in wanting to sleep. It means getting fed up with something.
"The boy begins to become tired" works too.
I wrote "pojken är att börja trött." I got this wrong. Why?