Hi! I've seen lots of great pallet coops that are quite sturdy, so that's definitely an optiom. As for the run, if you're worried about dogs, use 1/2 inch hardware cloth on the bottom few feet of the run. It's expensive, but dogs could easily rip through chicken wire, so the extra protection is necessary. Above that, if you'd like to save money, you could maybe use chicken wire or welded wire AS LONG AS the chickens are locked up securely in their coop each and every night. For dogs, you will also want to either bury welded wire into the ground around the run, or create an apron, to prevent dogs from burrowing in. More info here: http://www.mypetchicken.com/backyard-chickens/chicken-help/What-is-an-apron-fence-and-why-is-it-helpful-H282.aspx
Well, getting baby chicks will be significantly less expensive than buying juvenile started pullets. Plus, babies are super cute, and you'll be able to tame tbem most effectively if you start handling them as chicks. Then again, you'll have to wait 4-6 months for eggs, whereas depending on where you get started pullets, you may only have to wait a month or two. So that's up to you.
You certainly don't need a rooster. Your hens will lay eggs regardless. Reasons you might want one would be if you want fertile eggs so you can hatch at home, or if you free range and want a rooster to watch out for your flock. If neither of those apply for you, then you don't need a rooster.
To hold 10 chickens, you want a minimum of a 40 square foot coop and a 100 square foot run. If you ever think you might add on to your flock, I'd build bigger. You're better off doing it right the first time than having to change things later! Here's a great article on the space chickens need:
The cheapest feed will likely be at a local mill that makes their own feed. For example, in my area, the cheapest commercial feed at my feed store (Tractor Supply) is $12.99, and that brand arguably isn't that great. But at a feed mill a few towns over, they sell really good feed for $11 something. To find local feed mills, ask other poultry keepers...they're the ones who would know. You could also try asking feed store employees, although they might not know.
My Pet Chicken actually does sell Golden Comets...I think they call them Red Stars. They're the same breed, as far as I know. Easter Eggers are great, but if you live in a cold climate, they pretty much stop laying during the winter...or so I've heard. Climate matters a lot when choosing breeds, so we can give you better suggestions if you tell us that.
Oh, and one thing to keep in mind as you're drawing up plans for a coop is ventilation. Fantastic article here:
Hope that helps! Tell us your climate and we can help more.