I would suggest that you go to the Backyard Chicken Message Board & read a bunch of the posts there. You can also ask them lots of questions.
You need more that just a 'yard' for chickens.

I see no reason why it wouldn't work, simplistically. In the woods I'd be worried about predators-owls, coyotes, foxes that will snatch those chickens up quick unless you have some sort of plan to deter them. Would they be clipped?

Are the pine trees growing in sandy soil? If so & it's dry the area should work for both goats & chickens. If it's a dark, rich soil it may be too wet/moist. What are you planning on using for shelter?

If the area is large enough the grass area would be beneficial. If not, the grass will die from all the scratching & trampling. Maybe it would fare better if you layed down some wire to protect the grass, yet they could still enjoy it. Dunno if that would work as I never tried it.

Goats & chickens can live peacably, no health issues between the species, unless the area became too wet.

I'd be worried about predators too; especially racoons and birds of prey. Goats love the sun and would be unhappy in a shady spot. Chickens need the shade in warm climates. Grass is good for both. They definitely need a shelter of some kind IMHO

I agree with gakaren - visit the Back Yard Chickens forum where you will get a lot of great information and ideas. I mean alot of good information!

Here is a link that might be useful: Back Yard Chickens

I wouldn't plan on using the soiled pine straw to improve to look of your flower beds.