Do I have to finish playing this dlc by July 8? And if I don’t, will I have to purchase this add on even though I received it “free,”iIn order to finish it?
As far as I understand, no. Once you’ve purchased it, you will have it forever.
I think it’s free for a month possibly to generate more hype for Borderlands 3.
The Handsome Collection is also free on PS for PS Plus users, so they may have wanted to keep it consistent with the rest of the game.