Is PMI required only when when taking out a mortgage that's 100% of purchase price or when you put down less than 20% as a down payment???

Generally its required on all loans that are less than 20%, although some banks will allow you to go without with a smaller % down. That being said, for all new loans, PMI is now tax deductible. So it really is not nearly as painful to pay as it used to be. Going with an 80-20 loan is far less advantageous than it used to be as well.

PMI is required by Fannie Mae/Freddie Mac any time your first mortgage exceeds 80% loan to value. It doesnt matter if it is a purchse or a refinance. Any conventional loan over 80% ltv comes with pmi.

There are ways around it. There is what is called Lender paid mortgage insurance. Depending on the loan to value you can pay a little higher rate to avoid having the PMI on your payment every month. There is also a lump sum PMI where it is paid all up front.

PMI is also tax deductible assuming your household income is less than 100k a year.