'Can he not read either?' is actually perfectly acceptable English, if a bit formal. You'd just have to separate the negating word from the (modal) verb. 'Cannot he read either' does indeed sound very old-fashioned, and a bit poetic perhaps ... it might sound OK in Shakeaspeake, for example. Only about 500 years out. :-)
It sounds very unnatural but would technically be correct. Just as in English we'd say "let's party!" And is sounds strange to say "let us party!" The first sounds like a suggestion or command and the second sounds more like a plea but could also be the same as the first. It just sounds so unnatural because it's old-timey and rarely, if ever, used.