J initially believes that testosterone shots alone – and maybe surgery later on to complete his transformation into a man – will “cure” most of his problems. Does this seem likely? By the end of the novel, does it seem as if J still thinks so?

It is likely that after J takes his testosterone shots and undergoes surgery, his problems would be solved. The main cause of his frustration is people misidentifying him as a girl, including his parents and closest friend. In addition to that, he faces stigma from people referring to him as a lesbian when he is not. 
J knows he is a boy trapped in a girl’s body, but after testosterone shots and surgery, he hopes to attain a male look, including a beard and a deep voice.  The intervention would finally eliminate misidentification and pave way for acceptance by those around him.
By the end of the book, J still has the same sentiments. He enrolls in a gay school and makes two new friends, Chanelle and Zak, both transgender. He is pleased when he receives an acceptance letter from college addressing him as Mr. Silver.