Must this hash value start with a specific number of zeros
In a sense, but that is not entirely accurate. The hash must be smaller than the current target. This is not the same as having a specific number of leading zeroes. For example, 0x00001FFF and 0x0000FFFF both have the same number of leading zeroes, but there is a huge difference between those two values. (The first number is 8,191 while the second is 65,535.) If the target was 0x0000CFFF, then one of those numbers would pass and the other would fail. (The real target is a 32 byte number, not a 4 byte number as I have used here.)
With this can I say that hash rate is directly proportional to a specific miner or pools profitablilty
Essentially, yes. The higher proportion of the hashrate that a specific miner has, then they will find a higher proportion of all blocks, and therefore earn a higher proportion of all block rewards.
With the hash rate of Bitcoin hitting 1,000 Th/s in 2013 can miners use mining hardwares other than ASIC miners?
They can, but they will be incredibly unlikely to ever mine a block successfully, or if they contribute to a pool, they will earn a tiny fraction of the total rewards.