There are too many answers without a clear and correct answer. Many companies do binning on their parts so they close a conservative hold corner( Fast models, high temp etc). But setup they close on TT, and then sell the parts depending on what frequency they work at!!.... so if the part works at 2Ghz they sell at that speed and if works at 1.8Ghz....
Typ corner is run for power estimation for the part ( for customer)....assuming as the process matures it will come close to TT and that power is representative power.
With the complexity in designs you can have violations at TT corner as well.... but yes running TT corner is usually a low priority and is done before the design is finalized.....