With only one winning block (generally), yes your chance of mining the specific winning block doubles.

There is no explicit co-ordination regarding blocks mined, and no hashes tried by other miners have any impact on you or your odds, except if they win, as there is no (known) co-ordination between (independent) miners on what has or has not been tried yet - any block which produces a valid next-in-chain value is the 'right' one, and every miner who gets a value which isn't the right one, got a wrong one.