You can't investigate every claim on the internet in great detail, so dismissing the more unbelievable ones out of hand is not a bad strategy. But you need some kind of reason. Either a known bias or untrustworthiness of the author, or knowledge that at least some of the claims made are false.

No, if you can be right about numbers not adding up without being able to point to an explicit reason.

But it will harm a child. This has been shown often.

It has been shown often that homeopathy works and that telepathy works. That doesn't mean it makes sense to believe that it works.

I'm not aware of randomized controlled trials for occasional spanking and it's likely one of those shared enviroment effects where we know from twin-studies that they don't have much effect.