"Is it living?"
Hmm, "why does it mean 'living'?"... The word levend is constructed in the same way as dansend, lopend, schrijvend (=dancing, walking, writing), i.e. by adding -d to the infinitives dansen, lopen, schrijven (=to dance, to walk, to write). As such, these words can be used as adjectives and/or adverbs, e.g. "de dansende vrouw; Lopend kwam hij naar het station; een schrijvende man". So, een stervend paard = a dying horse; een levend paard = a living horse.
Also: I really think the most natural translation for "Leeft het?" is "Is it alive?"
The infinitive form is leven.
"Is het aan het leven?" is grammatically correct, but it sounds strange to a native speaker. The continuous construction in Dutch ("aan het" + infinitive) is usually reserved for actions that are going on as you speak. Phrasing it this way implies that living is an action, which, to me at least, sounds odd. Perhaps someone else can explain it better.
As a native English speaker, I remember "aan het" by analogy with "on the". It's not a particularly common form of the continuous in English, but it's sometimes used for emphasis. For example, "the lions are on the hunt" implies that you shouldn't go outside because they might eat you, whereas "the lions are hunting" sounds a bit less ominous.
As it applies to Dutch:
"Ik ben aan het lopen" -> "I am on the walk", which sounds a bit unusual but makes perfect sense to me.
"Is het aan het leven?" -> "Is it on the live?", which doesn't make sense at all.
This is far from scientific, but it's the way I remember things. Thought it might be helpful. :)