Not "can" but "might be able to". For example, new builds might get you a lower interest rate, but if you go to refinance, then it's not a new build anymore. (Though in one way this is usually true - a lower principal will give you a lower interest rate usually)
A lot of banks do charge you a penalty for repaying before the (fixed interest) term in case of fixed rate mortgages.
Not in the USA. Any mortgage allows repayment without penalty. Banks actually want it that way as their business is underwriting loans and the fees associated with them.
