Can I be sued if my dog injures somebody on my property?

Your homeowners should cover this and it was an acident the dog did not attack or maul the child it slipped and so did the 2 year old. By the way this is why you also need a 1-5 million dollar umbrella policy. Sounds expensive it is not bad at all.

PS this was an accident!

Quite honestlly the breed in this scenario is irrelevant.

1) can I be held liable or sued if our friends' daughter has sustained any kind of serious injury?

Sure you can, just like you can be held liable for anything that happens on your property. Obviously the dog didn't attack the child, rather he was running, took a slide and took the kid out like a runner heading for home takes out the cather!!

2) Does having a "black-listed" Doberman have any influence on that?

Not sure what you mean by 'black-listed'. If it is a breed that is not authorized in your area, then yeah, it will certainly play a role. If you use the term 'black-listed' simply because you think people are out to get the breed, then no, the breed is irrelevant.

Yes, you can be held accountable in court for injury done by your dog. Check your homeowner's insurance to see if you have coverage for this type of loss.

There is another issue. In many cities, the law requires that, if a dog attacks a human, the court will decide if the dog must be put down or ordered removed from the city limits.

Not just that breed but any type of accident on your property can fall back on you, but I would imagine your houshold insurance will have a personal injury clause to pay any bills, medical or other wise.

Yes.

yes

let your insurance take care of the medical bills

Yes, and the court can decide if your dog should be put down.

Yes, Yes you can