Living can be an adjective, but its usage is I think quite narrow. "Living creatures", "living quarters", used of extant languages, poetic use ("living flame"), and in some idioms such as "in living memory", "he is living proof of ..." etc. I don't think it would be correct to just use it as a synonym of "alive".
Living has to come before a noun when used as an adjective. "A living creature" and "the creature is alive". You can't say "an alive creature", neither can you say "the creature is living(as an adjective)". Though the latter phrase can be used with "is living" being a verb form, it doesn't seem to be an accurate translation of QUIERO LO VIVO.
Yes, but answer me this: “Is he alive or is he dead?”
See? He is lying on the ground there and looks dead and I want to know whether he is alive or dead. I need to know this because the guy owes me money. So I just asked the above question.
There are only five possible legitimate answers in English to my question.
“He is dead.”
“Dead." (Short version.)
“He is alive.”
“Alive.” (Short version.)
“I don't know!”
"Vivo" is one of those adjectives that changes meaning when used with ser versus when it's used with estar. I guess a way to think of it is 'to be lively/sharp' is a personality characteristic, so used with ser, and to be alive is a state. This page has a chart of some of those words: http://www.spanishdict.com/answers/100040/ser-vs.-estar#.VVvkf1VViko
We had a related conversation a few months ago, above. But basically, vivo is an adjective here, so it has to match gender/number with the subject. If it were "she is alive today", it would be "Ella está viva hoy". Or for "they", it could be "Ellos están vivos hoy" or "Ellas están vivas hoy".