Did The Who write their own songs

Originally they did not, eventually they sang and played themselves. They had a backing band when they toured, but did play and sing:

"In assigning instruments for purposes of the television show, a dilemma arose as none of the four was a drummer. Both Nesmith, a skilled guitarist and bassist, and Tork, who could play several stringed and keyboard instruments, declined to give the drum set a try. Jones tested well initially as a novice drummer, but the camera could barely capture him behind the drums because of his short stature. Thus, Dolenz (who only knew how to play the guitar) was assigned to become the drummer. Tork taught Dolenz his first few beats on the drums, enough for him to fake his way through filming, but Micky was soon taught how to play properly.[12] Thus, the lineup for the TV show most frequently featured Nesmith on guitar, Tork on bass, Dolenz on drums, and Jones as a frontman/singer/percussionist."

"The massive success of the series and its spin-off records created intense pressure to mount a touring version of the group. Against the initial wishes of the producers, Dolenz, Jones, Nesmith, and Tork went out on the road and made their debut live performance in December 1966"

"With Jones sticking primarily to vocals and tambourine (except when filling in on the drums when Dolenz came forward to sing a lead vocal), the Monkees' live act constituted a classic power trio of electric guitar, electric bass, and drums (except when Tork passed the bass part to Jones or one of the Sundowners in order to take up the banjo or electric keyboards)."