Let's start out with a question.
Should you use treats for training a dog?
The answer is yes.
Because it helps lure the dog, into the cue you are wanting it to do. Such as, a sit, or down. Then once they do the cue all the way you reward them with a treat.
Should you always use treats for teaching your dog something new?
Again, the answer is yes. It helps shape the cue and the treat is rewarding to the dog.
Should you always use treats, every time you want the dog to sit or lay down? Even though it already knows how to do it?
No you don't need too always give a treat every time you have your dog sit or lay down. It's always good to reward them for these cues though.
Let's say you have the dog doing push up's, which is sit, down and stand. For five different times. If your dog already knows how to do them, reward with a treat on the third and fifth time.
When you move to a new area to do dog training, with your dog make sure you have treats. Because the dog may be a little nervous, and needs some encouragement.
Do I need to always carry treats with me when I walk my dog?
At first yes you need too. You also want to have treats with you, incase your dog gets distracted by someone else walking, or a squirrel. Personally I always carry treats with me, to reward the dog, when the leash is loose when walking. And also to help regain the dogs attention, when there are distractions.
It is always good to reward good behaviors that your dog does. :-)
You can always take some of your dogs food to use as treats.