Can goku turn ssj5 in Dragon Ball Z?

Answer 1

No. In DB-Z, Goku can only go as high as SSJ3. He learns SSJ4 in DB-GT, but that's all. The rumored DB-AF (AF means April Fools) has been debunked.

There is no such thing as DB-AF and therefore no such thing as SSJ5 except in the minds of those stubborn enough to still believe it exists. There has been almost no new DB material released since the end of GT and most of what was released was recompilations of old stories. Akira Toriyama himself has said that there will be no new DB series released and that GT was the end.