How long does it take to build an FPGA?
I'm assuming you mean "fabricate" here, rather than build.
The time to fabricate an FPGA is irrelevant to the user in most cases.
Since they're all fabricated the same, you can use one available from stock at your distributor or at the factory.
In case the factory runs out of stock, it may take 12 to 16 weeks to produce more.
How long does it take to build an ASIC?
I am not an ASIC user, but I guess this is in the range of 12-16 weeks.
How much does it cost to build an FPGA?
Again this is irrelevant to the users. What matters is what it costs to buy an FPGA. This can range from $5 to $5000 depending on the amount of resources in the FPGA.
How much does it cost to build an ASIC?
For the same function, this will be lower than for an FPGA on a per unit basis. However you will have to pay non-recurring engineering (NRE) costs to set up to fabricate a particular ASIC and these might be several hundred thousand dollars or more.
How long does it take to program an FPGA?
If you mean, to create the design for the FPGA, this depends on your skill and the complexity of the design.
If you mean, having created the design, how long to put it into the FPGA, this is typically a few seconds.
How long does it take to program an ASIC?
ASICs aren't programmed. They come from the factory configured to do the function they were designed for.
Note: The word ASIC gets used for many different things. I am assuming you are talking about a traditional "structured ASIC" which has a fixed selection of gates available in the the silicon layers and is configured for a particular function by changing the metal layers that interconnect the gates.
The word "ASIC" is also sometimes used (at least informally in my experience) to refer to fully-custom IC designs, where all layers are free to be modified by the customer. These would require more design effort and higher NRE costs than a structured ASIC.
There are also services that take an FPGA design and convert it to a factory-fixed design (rather than field programmable). I haven't used these services, but I would guess they have lower NRE but higher unit costs than structured ASIC products.