Band of Misfits copying Feast and then Island
As of September 2019, Band of Misfits has received official errata that changes its behavior in regards to this question.

The new text for Band of Misfits is:


  Play a non-Command Action card from the Supply that costs less than this, leaving it there.


So Band of Misfits no longer "becomes" another card, instead each time you play it, it causes you to play another card from the supply without that card moving from the supply.

This means that when you use Band of Misfits to play a Feast, the Band of Misfits is no longer trashed. And neither is the Feast because the effect expects Feast to be in play but it isn't. And when you use it to play an Island, you will set aside a card from your hand onto the Island Mat, but neither the Band of Misfits nor the Island will get set aside.

So the new result of Throne Room + Band of Misfits choosing Feast and then Island is:


Play Throne Room 


Play Band of Misfits


Play a Feast from the Supply


Gain a card costing up to $5


Play Band of Misfits (again)


Play an Island from the Supply


Choose a card from your hand to put on your Island Mat