You never leave off the final "e" in the French indefinite articles (un, une), as you do with the definite articles (le, la, l'). Although, come to think of it, I can't imagine why not... unless it's because when you leave off the final "e" it looks like the masculine article.
A more specific answer is that the "e" in le is pronounced as an actual vowel, and thus it merges with the nearest vowel sound in the noun (e.g., l'homme), but the "e" in une is only there to denasalize the "n", creating a consonant sound where there is none in un. That "e" isn't really a vowel, so it must remain part of the article and can't merge with other vowel sounds.
Not directly, but the two words are probably cognate. From Wiktionary: couper comes from Old French coper, colper (“to cut off”), probably, derived from cop (“blow”), colp (modern coup), with its meaning coming from the idea of cutting off with a blow. But the Modern French coup (blow, strike) does not come from Modern French couper (to cut), no.
My translation "He is slicing an orange" was marked as wrong, although when I checked here http://www.wordreference.com/fren/couper, there is indeed a translation of 'couper' to 'to slice'. I am not a native English speaker, but I have never heard the expression "cutting a fruit". Am I wrong?
I would guess nothing is wrong, and without context either could be right. I am assuming here that in French the verbs to cut up and to cut both translate to couper.
Would be interested to have native french speaker expand the translation of to cup up, maybe couper en morceaux, and faire une seule coupe for to cut?
Unfortunately, you just have to memorize the gender with every new noun you learn until the associations are second nature. Some people give a simple rule of "nouns ending in '-e' are often feminine", but there are so many exceptions to the rule that it's not worth thinking about.