You'll also break a design down into units or blocks. These can be tested separately. E.g. you might see an L1 cache as a single unit and have a testbench (piece of HDL to stimulate and check a design) to test only that. This is useful as it's far quicker than a full design test however extensive full design testing is still required as many bugs you see as the consequence of the interactions between units.
Once you've done a lot of testing you'll tape out a chip. This may well have some issues so you'll do a new 'spin'.
As for testing for correctness, ASIC designs are almost always tested in software simulation instead of on FPGAs. Mainly because RTL written for FPGAs can look quite different from RTL written for ASIC synthesis. There's also a lot incidental engineering work needed to get something working on an FPGA.
It's actually very common to test ASIC RTL on FPGA. You are right that if you are targeting an FPGA you may write things differently, but this doesn't mean RTL written for an ASIC won't work it's just it might not clock as fast as it could and may not make the most efficient use of resources.
There is a lot of engineering work to take an ASIC targeted IP and run it in an FPGA, but thorough verification of an ASIC design is extremely important so it's usually worth the time to bring up an FPGA that can run it (sometimes as it's too large you have to split it across multiple FPGAs).
Especially if you try to make it go faster