Guest Joe F - I'm in the same situation; having learned Russian at school I have forgotten a lot, but I'm able to use a dictionary, too. This song I know by heart from my boy scout times on, and your translation is correct.
Only: Kalinka here is definitely a pet name for a girl, and might be derived from the tree rather than from a fish (which might be possible for the smell, but not for tender fondling).

Chey (engl. hey) seems a little bit unusual for my ears. While the Ukrainians are able to pronounce the H the Russians are not. So they pronounce it G, and the interjection gey! I often found in Russian songs (e.g. song of the Red Cavalry, about 1918), seldom ey!

Ay lyuli ... is in my songbook: Ay-da lyuli ... which fits the metre better. Lyuli is an onomatopoetical sound used in lullabies, and wandered from Slavic languages to Yiddish lullabies (e. g. lyulinke, may kind)