What is wrong with the construction "Он отдыхает сейчас?"
Adverbs normally come before the verb in Russian
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.