Do I have to set my hand aside so that card doesn’t mix with it, and play that card exactly, or does this just add to my total play, so I can add the extra card to my hand, and play whatever cards I want, increasing the number of plays I have?

A: “Draw and Play 1 extra card per turn” just increases your total draws and total plays for the turn by 1, so whatever card you draw that is “extra” just gets added to your hand like all the other cards you drew, and you can play whatever cards you want from your hand up to the play quantity plus 1.

Unless the card says you have to set aside your hand, you don’t, and you’re not required to play the specific card you drew card unless the option you’re invoking specifies that.