More precisely: Does the cost of making something have anything to do with the price it is sold at?
That said, it's by no means a floor, particularly on goods that stick around a while, since the things already in existence can more than meet the demand, which can drive the price arbitrarily low.
Not really. You can't recover the costs by "melting down" bitcoins, so it doesn't provide a support for the value of bitcoins the way that the use-value of gold might provide a support for the value of gold currency.
Clearly, if the cost to mine coins is less than the market value plus the expected transaction fees that can be earned by mining (mining is also transaction verification), there is no economic incentive to mine, but that doesn't operate as a price floor so much as a limit on the viability of the system.
As the value of the created coins go up, miners can afford to spend more on the effort in an attempt to capture a bigger share.