Well, it'd be almost impossible to determines if a 'sign from God' was actually legitimate- but I expect that at least some cases could be assumed as illegitimate due to their content, with just a little bit of scrutiny and asking the question "Is this consistent with the concept I have of God, as taught by my religion?"

I'd say a case like "Kill your son and then yourself" is not. There are a couple of cases in the Bible (Isaac, Jesus) where a father either sacrifices or is asked to sacrifice his son- but in the case of the former, it was a test of faith and redacted before the actual act was committed, and in the case of the latter, it was God himself, ostensibly, making the sacrifice (not that it turned out to be much of a sacrifice given, you know, resurrection). But a simple killing, shot to the back of the head, no ceremony, no sacrifice? Not much of a precedent for that- much less suicide immediately afterwards.

Let's not forget that God is not the only thing that can put voices or images in ones head- numerous mental conditions can also cause this, and if the christian mythos is assumed to exist, the devil is famous for his deceptive whispering as well. "Send your son to heaven and yourself to hell" would be far more consistent with either mental illness or the influence of the devil than it would be with the christian god.

why wouldn't even a Christian believe this woman who killed her son so that he could go to heaven?

Ever notice how every christian claims that they have the right interpretation of some verse, and all other christians who believe otherwise are just not annointed with gods spirit like they are?

Same concept... the woman claimed god told her to, but other christians object because the little voices in their head don't match, and since every other christian is the self proclaimed arbiter of whats inspired by god they can profess what is god vs just plain bat shit crazy.

You test the Spirits to see if they are from God.

No one can prove it. It is strictly in the mind of the beholder.