New bottle is better. You could probably give your child the Benadryl (diphenhydramine) without any adverse side effects. Perhaps the potency has decreased but if the medicine was properly stored, it shouldn't cause any harm. But if the medicine has been expired for two months, it's probably been in your cupboard for over two years. If your child had a bad allergic reaction, you'd feel a lot better knowing the medicine was still good.
No problem. With that timeline I don't think a dose or two would be bad in the short term. But put it on your shopping list to get the next time you are out.