I can tell you about our greyhound but you should also call a local greyhound rescue and ask those questions.
She definitely alerts and can curl up into a very small ball at the foot of the bed. Mine has really no prey drive and is fine with an 8 lb wiggly chi that just landed here a few weeks ago.
I worry about bloat and feed her to avoid it.
Was yours a young rescue? It’s my understanding that those that are adopted out early are usually those that don’t want to run. Maybe that indicates a weaker prey drive?
What food do you feed yours?