"He is now having a rest."
Translation:Он сейчас отдыхает.
Why doesn't "having" signal the "у ему" construction. "Она сейчас отдыхает" seems like it should translate to "he is resting now".
There's no need for "у него есть" construction. The verb "to have" doesn't mean possession or ownership of something in this sentence.
To have a rest = to rest
He's having a rest now. = He's resting now.