Yep, this sounds fine to me. However, in your question, I think you mean to ask is it OK to give some*thing*, not some*one* - a dog is a thing, not a person.

yes this sounds perfectly fine to me :) Although if I was talking about my dog I would probably refer to the dog by it's gender. So for example if someone asked me; Who are you going to give your dog to? I would say: I'm going to give HER to Jane. If it is already known that the subject is a dog it would not be necessary to repeat the word dog. Otherwise yes that sentence is good.

This sentence is grammatically correct. It's possible to "give something to someone." You could also say, "I'm going to give Jane the dog." As long as the object being given and the person to whom the object is given are clear, the sentence will make sense.

Your question is not clear to me You cannot GIVE a person to another person literally. (Well, perhaps in some countries, you might be able to but not in free western countries.) You can HAND a baby to its grandma. You can PRESENT someone to someone by way of introduction.