Cocktail & Burger on Sauchiehall comes the closest to In n Out, in my opinion. It's a sit down restaurant though, no fast food places do good American style burgers here.
Not familiar with On the Border but we have two Chipotle style burrito places - Taco Mazama and Pinto's. I think Taco Mazama is the best. Chiquito's at the Kingston Quay is a more sit down Mexican restaurant. It's a chain so has that Chilli's feel with chips and salsa and margaritas. They only give you a few though and last time I was there we got popcorn (?!) when we sat down instead of tortilla chips. And the Mexican food is just ok.
The only hash browns I've had here have been at McDonalds! TriBeCa is really good though. Massive portions. Not sure what you mean by American sausage? Like more of a seasoned link sausage? I think the British link sausages don't have enough flavour.
Not really had much BBQ food here but I've heard good things about Smoak and Buddy's.