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.
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").
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".