What roles has the TIME information play in the bitcoin header? What role does the time play in the bitcoin header? Is it ok to set it to 1 sec after the last block and never change it while calculating? Will the block be rejected when the gap is to large?

If 2 people get the correct block ( reach the traget ) who will win? Does the time decide?

Information about the header is taken form here:
https://en.bitcoin.it/wiki/Block_hashing_algorithm
the time feature really doesn't play a role in the broadcasting of two simultaneously mined blocks. Those are just handled through broadcasting effect, forking and subsequent blocks mined. Time is started from the moment the miner began hashing to solve the POW. Its used to know how long it took in comparison with recent blocks found for the difficulty re-targeting calculation. It also serves as a record to know when a block was mined.