It gets the idea across OK, but I think the other order ("Those who are hungry, eat") is much more natural in English.
And the Hungarian one? Azok akik éhesek, esznek? This would be the same, like the english one.
I think it depends on which logic you want to follow. Do you want to explain why the people eat? Then you'd say "Azok esznek, akik éhesek." If you want to say instead what hungry people are doing, you can go for "Azok, akik éhesek, esznek." Compare with German: "Es essen die, die hungrig sind" vs. "Die, die hungrig sind, essen." Both are totally valid, the first version just sounds a little better.
I think "Those eat who are hungry" should be accepted, too, even if it sounds a little troubled. I'm not a big fan of subordinate clauses in the middle of the sentence anyway.
"Those are eating who are hungry" seems OK, and how about "Those who are hungry are eating"? ...
Sure. For all intents and purposes those sentences are the same. The second sentence sounds a bit better, though.