Silly questions..

Is BM needed in dungeon runs like TT and HH ? If yes , why?
("Twilight Temple" and "Holy Hall" are the same thing, TT is just the International name for it.)
-I always prefer to run with at least one BM in party. Defense buff aside, they're good at catching stray mobs (or ones from a pull that run too fast), killing bosses faster, and they tend to be more 'fun' to be around. Something I call "BM's Disease" - an almost reckless habit to charge in swords/axes/fists blazing - makes life a bit hairy and dangerous, but adds to the thrill as well. TT runs get old and repetitious after a while...

Which class is better in pvp?
Depends on the build, and who you're facing. BMs have the best bonuses for accuracy/evasion per Dex, so a high-dex BM is nearly untouchable in melee. Combined with their ability to stun-lock... well...
Barbs on the other hand have ridiculous HP, especially the tank builds. You won't kill as often, but you'll shrug off damage like nobody's business.
Personally, I don't PvP, so other opinions might be more valid than mine.

Who will win when a BM vs a Barbarian ( same level , same equipments 1vs1)?
Again, depends mostly on build. BMs make you a sitting duck with stun, and recover HP with Sutra. Barbs have Sunder to recover HP WHILE doing bleed damage, and higher HP reserves. But both have minimal magic reserves - generally speaking, the longer the battle drags on, the more it favors the barb.

Which class is more efficient in pvp 1 vs 1 ( stunlocking BM & barbarian) to kill range classes?
If the BM can sprint there fast enough, or get the drop on a ranged class, I'd say they were more efficient at the kill. Barb damage tends to be lower, thus a slower kill. We're built to last, not assassinate.
THAT SAID, using Tree of Life (genie) to super-inflate your HP reserve and dropping Perdition (Armageddon) on a target will deal out some VERY hefty damage, and from what I gather is a popular tactic for nuking squishy types. As long as you don't miss.

Is barbarian able to grind alone fast at later level? Or they only could tank all the time to earn exp?
Yes, albeit not [as] fast as a BM. Some barbs I've seen have a cleric accompany them and share the exp while dishing out Slam, Swell, Firestorm, etc. My own personal favorite tactic is to gather up a crowd and use Sunder/Surf Impact to AOE grind. It's not as quick as a BM's AOE grinding, but doing so in tiger form means taking less damage, less charm burn, less risk of dying.
And if you visit any of the popular AOE spots you're bound to see a dead BM or two lying around...
Of course, there's always Zhen. Zhen parties like a good barb for luring, though admittedly I'm not particularly good at it.

Which class consumes more hp charm in pvp?
Don't PvP, couldn't say. But given that Barbs are tanks and meant to take the punishment... well, I'm inclined to say Barb.

Which class(BM & barbarian) has higher chance to kill wizard at lvl 90 ++?
LA Bowbarian. b:chuckle They'll never see it coming.
Honestly, probably the Barb. Providing you can reach and hit the wizard before they nuke you off the map, you stand a better chance of surviving to finish them off. Plus, Alacrity pounce will cancel a cast - and wizard casting is kinda slow.

What does rage of the beast king do?
Beastial Rage - a red circular icon?
When engaged in combat it helps build up Chi - every hit you take will add points to your chi meter. This is essential in tanking, since the most critical skills (Flesh Ream, Invoke Spirit, Sunder) all require chi/sparks.

If BMs arent popular in TT dungeons , does that means they would have a harder and longer time to earn money / Mats for their TT equipments?
Not if they're proficient at AOE grinding. Had a 90+ BM tell me she earned over 450k [after repairs] in a few hours. In the same span, I'd be lucky to break 150k - and my repair bill would cost a third of that.

some barbs like Kephras believe that it is essential for tanking while it's kinda useless since most bosses and mobs u have to tank hits kinda slow and u don't recharge much chi... u really understand it's power when u do small aoe group parties or inside rebirths when u have 50 mobs on u

Fair enough, a lot of bosses do have a pretty slow hit-rate (Serpentrex in '69 hits pretty fast though). Personally I consider it essential because it provides the extra chi to Flesh Ream, Devour, and still build up enough spark to use Sunder or Invoke as I like. Without, it's more difficult to do so - venos can give you sparks, but won't always do so and it's best not to rely on them for it. Same for apoth. chi powders - long cooldown won't help you except in an emergency.

EDIT:

...and no, barbs can't AoE grind, BM's are the kings of solo-aoe grinding

I won't dispute the title, but "no, we can't"? >:} Watch me, Kosake. Been doing it on Nightscream for the last week or two.
-We CAN AOE grind, we're just less efficient at it. We're also less likely to end up dead for it.

Surf I'll spam, Alacrity takes timing (I maxed mine). Frighten, I've never seen much use for. But, to each their own - I certainly don't have a problem keeping up chi, regardless.

did u really maxed Alacrity ? bad choice

Not really. I've explained in other threads (clicky!) why it's not a cardinal sin to do so - it just goes against popular opinion. But then, I've done a lot of things with my 'build' that might go against popular opinion - I'd prefer not to hijack the OP's thread with that discussion, though.