If a spell is countered by say, Dovescape, and Guile exiles it, if that card is played would it be put into the graveyard, or exiled entirely?

Someone with more rules knowledge feel free to correct me, but I believe that when you cast a spell from exile using Guile's ability it will follow the normal procedures for casting a spell. This means you move it to the stack from whatever zone it's currently in (usually your hand, but in this case from exile), pay the casting cost and any additional costs, and upon resolution put the cast card would into its owner's graveyard unless stated otherwise on the card (like Temporal Trespass or Beacon of Immortality) or because of some other effect (like Leyline of the Void. I believe the only difference between casting a spell "normally" vs. casting one from Guile is that the zone the spell starts from is exile rather than your hand.

Although, in the case of Dovescape in particular, if the Dovescape attempted to counter the spell the first time and the enchantment is still on the battlefield when you try to cast the same spell off of Guile again, Dovescape will attempt to counter it again, causing Guile to exile the spell again. Seems like a fine way to build up storm count and an arbitrarily large number of birds.

If you counter a spell while controlling Guile, you exile the spell and get an immediate choice: a) cast the spell now without paying its mana cost or b) leave it in exile indefinitely. If you cast it and it resolves, it goes wherever it would normally go after resolving (usually either the battlefield or the graveyard).

But yes, with Dovescape, it will presumably never resolve. It will get countered again, and you can then immediately cast it for free again.