What happens to price when the block reward is gone completely? What will happen, specifically to the price of Bitcoin and transaction fees when the block reward drops to a negligible amount?

Today, each block reward is approximately worth 25 * $300 = $7,500. To generate a tantamount reward as today, with (an assumed) 4000 transactions per block at today's exchange rate, the average transaction fee would have to be $1.87.
Under these assumptions that would put Bitcoin out of business in terms of being an affordable way to make everyday transactions, especially micropayments.

Now I know that block size is mutable so does this mean it's inevitable? Or does it mean that the price of BTC will have to rise to at least the cost to mine a block divided by the transactions per block multiplied by the exchange rate to be worthwhile? Doesn't this then mean that the value of Bitcoin outside it's functional value, is then ultimately determined by the block size which markets don't have control of? 

Does BTC fiat price increase to meet the collective cost of actually hashing the transaction block or does the total block transaction fee value determine it? 

My point is, when transaction fees are the ONLY incentive to mine (and that will happen eventually as a certainty) the total sum of the transaction fees must match the cost of mining the block they were included in. If the block costs more to mine than the value of the transaction fees in a block, miners (most) will shut down. At least so many will shut down that the network becomes insecure and Bitcoin will die.
You've completely failed to mention the difficulty retarget mechanism in your analysis. If the block reward drops, miners will quit until the difficulty is low enough that the remaining miners will break even. This will happen regardless of what the new reward per block is, the only thing that changes is the new difficulty/hashrate. (There is some subtlety in that the retarget mechanism takes time to kick in, but if the drop is gradual that's not much of a problem.)

With low hashrate, the network is less secure. This is why we need to come up with ways to make sure transaction fees are sufficient to fund an acceptable level of mining. (A limit on the data size of blocks is a possible such way, but not a very effective one IMO.)

But the drop in reward will not magically cause the price to go up. The price is determined by the market.