Does all miners in mining pool increase the nonce variable one by one starting from zero?
This is not how mining typically works in a pool. Mining software typically assigns a different extranonce to each miner, which is part of the coinbase transaction, and will result in a different merkle root (and different block header. Then they can all start on nonce 0 as you suggest, but they will not be racing each other for the same block because they are different. See https://bitcointalk.org/index.php?topic=9438.msg136344#msg136344.

Also, just for clarification, the data that is hashing during mining is the block header, which consists of:


Version
hashPrevBlock    
hashMerkleRoot   
Time     
Bits     
Nonce    


See https://en.bitcoin.it/wiki/Block_hashing_algorithm