is "pojken får trött." also correct?
No, it works in English but in Swedish that would be like saying "the boy receives tired" (or "the boy sheep tired").
I actually think "The boy becomes tired" works much better in English (I'm a native British English speaker), it's also much closer to the literal Swedish.
Would DL accept: -pojken tröttnar-?
I don't think so, it is not the same meaning in Swedish.
Bli trött=Become tired (and want to rest)
Tröttna=Become tired of something (or Get fed up)
However, "bli trött på xx" has the same meaning as "tröttna på xx"
It accepts "pojken tröttnar".