"Having a rest" isn't a natural phrase. "He's resting," or it's really implying a temporary rest during some longer task or at work, then "taking a break," would also be used.
"Having a rest", while not as common as the alternatives you mentioned, doesn't sound unnatural to me as a native British/American speaker. This could well be one of those things that depends on your dialect.
"To have a rest" is perfectly acceptable in Hiberno-English. It means "to take a break".
"Having a rest" is not common American English. Definitely UK English. There's nothing right or wrong about that, but you will be viewed as odd or affected if you use it where it's not usually heard. It's not heard on American TV for instance, except when we're watching BBC programmes. Fans of Downton Abbey might find it quite charming. I wouldn't use it in a redneck bar, though.
Never heard the phrase "having a rest" in the US, only in British English. Other variations on this should be accepted.
If he is cooking is accepting then he is resting should be accepted. Also when typing this in Russian I also got it wrong. I am not sure why seeing the word order is very flexible.
"отдыхать" (to rest) sounds like Polish "oddychać" (to breathe) can be a bit confusing to polish people learning Russian :)
Я думаю, что слово отдыхать произошло от слова дышать, поэтому звучит похоже. Шёл, стал тяжело дышать, остановился, отдохнул (восстановил дыхание).