Deceased parent last year tax return filing for 2020

Hello, 

 

  My father passed away on October 31, 2020. I was appointed by the court as administrator for his estate. 

  My dad is a resident of California and he has an accountant helps him doing income tax for 20 years.

  My dad paid estimated Federal and State tax for the first three quarters in year 2020.

 

  When he passed away, his accountant advised me not paying 4th quarter Federal and Stated tax for year 2020.

  1. My question is he lived through the month of October which is part of 4th quarter. Should I pay for 4th quarter tax for my father?

 

  2. My dad has an employer sponsoned 403b plan (403b plan held by Fidelity). He didn't take any distribution in year 2020. Will Fidelity notify IRS that my dad didn't take distribution from his 403b plan, hence he didn't need to pay any taxes for the 403b plan. Or Should I ask Fidelity mail me a letter to state my dad didn't take distribution in year 2020?

 

 Please give me some advise and thank you in advance!