Howdy Doody, all 28 inches of him, has come home to roost. On Friday, the handsome, freckle-faced marionette with the red hair assumes a prominent position on display at the Detroit Institute of Arts, where he will greet the public until May 13, when the Institute will decide upon his permanent display status. Howdy, who delighted ’50s children (and annoyed adults, because his peanut gallery was so noisy) until his NBC series left the air in 1960, did not have an easy time getting to Detroit, however, having been at the center of a bitter two-year custody fight between the museum and the estate of the late Rufus Rose, his puppeteer. In its legal battle, the museum had claimed that Rose, who took the puppet to his Waterford, Conn., studio after the program left the air, made the promise to donate it. Rose’s family, on the other hand, said there had been no such promise and, furthermore, refused to guarantee the authenticity of this Howdy. In January, a federal judge ruled that the puppet was the original and, therefore, belonged in the museum — where he will join a Kermit the Frog and a Civil War-era Punch and Judy in the museum’s collection of some 850 puppets.