Translation:Please cut off half of the cucumber.
This: "take the cucumber and cut up half of it"
"take the cucumber and cut it in half" would be "Разрежь, пожалуйста, огурец пополам (adverb)"
- отреза́ть (perfective is отре́зать) is about cutting and taking the cut part away
- разреза́ть (perfective is разре́зать) is about cutting itself
This sentence uses the perfective verb.
Yes. In usual American English idiom, the thing you "cut off" is smaller than what remains. If you cut something into halves, you "cut it in half" you don't "cut off half". If the thing you want is bigger than half, you simply ask for the amount you want - "Cut 3/4 of the [thing} for me". And you don't cut off the smaller portion and ask for the larger one. But, it's English, so there are always exceptions.