Mice are carriers of intestinal parasites, namely coccidia and roundworms, you'll need to routinely have them checked (dog and cat alike) for these parasites and treat accordingly.
These parasites also live in the environment so it's a good idea to check anyway.
I can't picture that there isn't at least one chomp to kill the prey, I've never seen a dog eat a living thing that eagerly, but you never know.
As far as stopping it, I see a few options.
An exterminator
Keeping the dog out of the conservatory
Work on the "leave it" command.
Personally I think I would opt for trying to get rid of the mice, sounds like there may be many if it is a daily event. If you do, do it safely you don't want to risk your pets by keeping dangerous rodentacides around, call an exterminator and enlist their help or advice.
As far as mice running around in the stomach, a normal healthy stomach should empty in about 2 hours, and stomach acid is strong stuff.
Edit: After some research I just found this. I'm not trying to scare you but it may interest you to also know that mice have been known to carry salmonella, typhus, and leptospirosis.
http://www.macroevolution.net/diseas...l#.Uk4AK9K-qWk