Saying "He put the shoes on backwards" isn't quite accurate in this context. Instead, you can say: "He put the shoes on the wrong feet." "He put his shoes on the wrong way around." These expressions more clearly convey that the right shoe is on the left foot and vice versa.

"Backwards" isn't quite right. In addition to Albert Lovell's answer, I'd add as a possibility, "He got his left and right shoes reversed." And another possibility is just to say "He put his right shoe on his left foot" and leave the rest unstated. An example of this occurs in a nonsense poem by Lewis Carroll's "Through the Looking-Glass:" "And now, if e'er by chance I put My fingers into glue, Or madly squeeze a right-hand foot Into a left-hand shoe, Or if I drop upon my toe..." He merely talks about putting the right foot into the left shoe. He doesn't say what he does with the left foot, it doesn't seem necessary. A very heavy weight, I weep, for it reminds me so Of that old man I used to know--

I wouldn't say backwards, that would mean the toe ends are on his heels and the heel ends are on his toes. I would say "on the wrong feet". Or "the wrong way round", which is less specific but most people will know what it means.

Yes. "Your shoes are on backwards."and "Your shoes are on the wrong feet." both mean the same thing, and are used interchangeably.

You can say: he put the shoes on the wrong feet.