7. After I (post) the letter I (remember) that I (forget) to put on a stamp.

After I (post) the letter I (remember) that I (forget) to put on a stamp.
After I posted the letter I remembered that I had forgotten to put on stamp
Shakespeare (be) born in 1564 and (die) in 1616. He (live) in Stratford for most of his life but he (live) in London when Queen Elizabeth (die) in 1603.
Shakespeare was born in 1564 and died in 1616. He had been living in Stratford for most of his life but he was living in London when Queen Elizabeth died in 1603.
Was born, died Past Simple Passive
Had been living Past Perfect Continuous
Was living Past Continuous
Died Past Simple Active
проверьте,пожалуйста, правильно ли

He LIVED in Stratford for most of his life
остальное вроде верно

Верно, кроме He had been living

After I posted the letter I remembered that I had forgotten to put on stamp. Все верно.
Shakespeare was born in 1564 and died in 1616. He had been living (had lived так же допустимо) in Stratford for most of his life but he was living (нужно lived, так как нет конкретного времени с точностью до часа) in London when Queen Elizabeth died in 1603.

 «died Past Simple Passive» какой же это Passive?

Времена названы правильно. Но в 1м я бы употребила еще Past Perfect вначале. И пропущен неопределённый артикль(надо «a stamp». Далее предложение должно быть, на мой взгляд, «Shakespeare lived in Stratford for most of his life but he was living in London when Queen Elizabeth died in 1603. Или::» Shakespeare had been living in Stratford for most of his life but he lived in London when Queen Elizabeth died in 1603. :))