I would suspect that the average greyhound would go for your rats or hamsters if you let them out in the vicinity of the dog, but if they are in cages, no. I had a dwarf rabbit when I got my first greyhound. She would look in the cage and even do nose-to-nose with my bunny. I would never dare let the rabbit run around with the dog in the vicinity. After all, they are trained to chase small mammals. But, with cages or whatever arrangement you have, it shouldn't be a problem.
With regard to your schnauzer, I would simply suggest that you introduce the greyhound and schnauzer in neutral territory and see how they get along. Our adoption group pre-screens dogs for "cat safe" qualities and with small dogs. Many are cat-safe and most are small-dog safe. A few are not. Good adoption groups will have homed the greyhounds with a family who has either cats or other dogs or small kids that can comment on the greyhound's behavior around their other pets.
The pool is probably no big deal. Some greys like water, others don't. Mine likes to get his feet wet on the beach of the lake, but won't go swimming. Other greyhounds I know love to swim.
I also live in the country on 8.5 acres. I have a fenced area that is simply fenced with 6 foot tall galvanized deer fencing. It comes on 50 foot rolls at the hardware store. We bought t-posts (the kind you use as tree stakes) and wired it to those. He runs around in there so we can let him out the back door without having to go out there ourselves in rain/snow/whatever. It is a little easier for us. We also have a 30 foot lead and he loves to run with us on that, but not for more than about 30 minutes because he gets tired. Like I said, they are not endurance dogs, although many greys can be trained to run for longer - but they are not used to that from the track. They only run every 3rd day.
Our greyhound is confused by other dogs that are not greyhounds. Around other greys, he is fine. Around other dog breeds, he is sometimes nervous. It seems as if he doesn't realize they are dogs? The other greys I had were not that way. If the labs around you are very rambunctious and playful, the greyhound may be initially nervous about that. They are only ever used to other greys. Many of them adapt perfectly well to other breeds of dogs though. Again, this is where a foster home is helpful in telling you such things.
With regard to sitting, they are able to sit (ours does so when he scratches his ears) but they are not extremely comfortable sitting like other dogs. If you see one lying down (not on its side) you will see that it is like a sphinx - its butt doesn't touch the ground. The front legs touch, and the chest, and then the back legs touch from the foot up to the "knee" on the back leg. The haunches of the dog are raised off the floor. This is why it is hard for them to be comfortable sitting. But, ours learned "Down" very quickly. He just does the sphinx laying down position. We make him do a down before every feeding time, when he is having a leash put on, etc. He is very good with routines. Most greyhound owners do "down" instead of "sit" because the dogs are generally more comfortable that way. It is not generally a problem unless you have some reason that you want the dog to sit instead of lie down.
Finally, yes there is a special collar. It is called the martingale collar. The reason is because their head shape means that they can easily slip out of regular collars. There are a lot of websites that sell really fancy collars. You can also get regular martingales at the pet store.
Here is a nice site for them - I have bought collars here and they are nice quality.
http://www.2houndsdesign.com/
Claire