yes

Yes, I would check with your home owners insurance also, because something like this could be covered. Most of the time ppl won't make you pay unless they are just being hateful. Friendly neighbors would understand that accidents happen no matter where they are. If you have some neighbors you would worry about I would either keep their kids out of your yard. It might be a good idea also to have a meeting with neighbors and come to an agreement and PUT IT IN WRITING with SIGNATURES just to be safe. It doesn't even have to be noterized if there are witnesses. If you did this than it would curve any conflict down the road! I would do it in a heart beat, luckily I am surrounded by family instead of just neighbors;)

You could be, but you are not automatically liable. Tort law is split into two categories: intentional torts and negligence. It is obvious why you should be liable for intentional harms and these harms do not relate to your question, so I will not address it here. So to determine if you are liable, the question becomes were you negligent? You are only liable for harm to another if you were negligent and that negligence causes harm. If you had a swing set (or pool for that matter) and kids entered your property without your knowledge and got injured, you would be liable in many jurisdictions. The reason for this is what is known as an attractive nuisance." We know that kids will use swing sets and pools and that they are not smart enough to appreciate the dangers in these items. Therefore, the law says you are negligent if you do not take steps to protect against this harm. However, if you do protect against this danger (e.g. putting up a fence and keeping it locked), then you are not liable so long as those steps were reasonable. If a kid is using the swing set properly under your supervision and gets hurt, you are not liable here. The reason for this is there was no negligence on your part. The important thing to know is that YOU ARE NOT AUTOMATICALLY RESPONSIBLE FOR ALL INURES THAT OCCUR ON YOUR PROPERTY. You are only liable when you intentionally cause harm, or when you are negligent.

Yes :)

yes, unsupervised children playing on your playground equipment. say the kid drowns in your wading pool. unsupervised swimming.

Yes. Your homeowners insurance should cover it.

Yes. If they're trespassing, you might be able to defend that. Your homeowner's insurance should include coverage for Medical Payments.