'I haven't much time.' Isn't it gramatically correct? A bit formal, but I am lazy, so I rather speak like a sir.
Grammatically and idiomatically correct. In American English, we'd say that, or "I have little time".
Even Duo's translation is a bit formal - we'd be more inclined to say "I don't have much time" or "I haven't got much time".
So, I guess this must be the positive expression of, yes, I have a little time now. How would one express the negative "sorry, I have little time right now"?
Actually, this one means it negatively: I have little time. I fixed the translation.
Много means "many, much, a lot", and мало means "few, little, not much".
Actually this sentence is more of the "I don't have time right now" kind. You would say «у меня есть немного времени» if you have some free time.
Hmm, that's why I wrote "I have little time" first, it was also my intuition that this was more on the negative side. In this case the suggested translation "I have a little time" is misleading if not even wrong.
As Norrius hinted a few days back, the difference between "мало времени" and "немного времени" is pretty much that between 'little time' and 'a little time'. The word "мало" conveys insufficiency, whilst the word "немного" doesn't.
Why it is времени instead of время? I don't understand this genetive thing I guess. How are people remembering this stuff? I've looked up the definition of genetive and such and I can't seem to understand it. So, Мало modifies time... and that's why time needs the genetive inflection?
Мало does modify время and this is exactly why you use the Genitive here. Use with amounts is just one of many, many scenarios where you need the Genitive in Russian.
A case is defined by its usage. In principle, you could just name them case 1, case 2, case 3 and so on, but the common approach is to give them names. Мало, много and other amounts will force the noun into the form that is (traditionally) called Genitive/ You might also use a noun with нет: whichever noun you choose, it should always take the Genitive when modified by нет. So one can say that the Genitive is the form that native speakers always use when a noun is paired with нет.
One of the things you should understand is that when a certain verb, preposition or structure requires a certain case, the interpretation does not matter much. The definiton of a certain case in Russian, or, well, a definition of what the Genitive or the Instrumental is in different languages, will help to a degree—but, in the end, it is all about the structures you know. If you know at least one structure where a certain case is required—congratulations, your knowledge of that case is not totally useless. Fortunately, you'll hardly ever run out of situations where the Genitive is required. A beginner should know at least three or four such situations and 5–10 prepositions triggering this case.
P.S. имя and время belong to a very small class of nouns. Their paradigm is unusual but manageable. The Nominative plural always ends in -а, and the pattern itself is similar to the one for кровать, дверь and so on (but you should insert -ен in all oblique cases; the Instrumenal ending is -ем)
Most quantity adverbs (e.g. много) take genitive plural, correct? Does мало take genitive singular?
PS: Google says "few onions" is «мало дука», but "few cats" is «мало кошек», and "few women" is «мало женщин». Inquiring minds want to know: what's going on? Does it depend on animacy?
Whether it is singular or plural depends on the noun.
The available time is uncountable, just like in English. If you did not have many names or cucumbers or horses it would use the genitive plural (У меня мало имён / огурцов / лошадей).