It is completely normal for a dog to eat grass. It helps if they have a upset stomach. My dog Scooby eats grass by water instead of drinking the water! He knows what kind of grasses are good and which ones aren't. I think he likes the taste and the moisture in the taller grasses.
Just like corn... grass doesn't digest very well. So yes you will see it in their stool. Mine often gag it up if they aren't feeling well. Especially the cat. They even sell special grasses for cats to eat at pet stores.
My dog Shaggy once pooed up a gum wrapper!!! So grass isn't that bad!