Do some Christian churches really believe that babies who die unbaptised go to hell? Would any sane god allow that?

Christianity in all its forms is an insane concept.

That is not to say any of the other religions are any better but Christianity claims to be a charitable, merciful religion and yet its church dogma includes the line "suffer the little children...."

It is a vile creed if it can espouse a belief like this one!

It has long outlived its use ( if it ever really had one) and now, like any sick animal, it needs putting down!

We are living in the 21st century!

We dont need this rubbish any longer!

Little children being punished for something their parents did not do.

What sort of sadists worked that one out?

There is not one verse in the Bible that states that

"babies who die unbaptised go to hell".

God does not allow that. Those are the words and teaching of men not God.

Some Protestant denominations believe just about anything you can imagine. After all, there are over 6,000 of them, the beliefs of each one contradicting the beliefs of the others. The original and true Christian Church, the one Church founded by Jesus Christ, to which He promised the fullness of God's truth, does not teach that God condemns to Hell children who have never committed a sin in their lifetime.

Yes, Rashers, some Christian churches DO believe that unbaptized babies go to hell.

As for your follow-on question, I don't think it makes any sense to try to apply "sane" or "insane" to gods.  It's a term invented for humans.  Angels and gods aren't just bigger, older, more powerful forms of human...If they exist, they are completely other from us.  "Sane" and "insane" don't apply, and wouldn't even have a framework for application.  They're ineffable.

No, but an insane god might.

Why do you even care?

What a load of nonsense!

For centuries people have lived " in fear of the lord"!

This benign, gentle God, who has terrorised the human race for millennia, has reached his "sell by" date and needs to go!

We are in the 21st century, this rubbish belongs 2 millennia ago 

There is a Biblical promise that God will save the children of the Righteous (His Elect,those who are saved). He made no such promise to the wicked and save their children or the babies they had aborted. I don't believe God would send those aborted babies to Hell but in His goodness,righteousness,justice and providence He would save them.

That’s why Christian theology is completely irrational. Islam says we’re all accountable for our own sins and will be rewarded or punished based on the deeds we did. 

Yes, there are some lunatic churches that believe that.

The Catholic church preached that unbaptized babies would be punished by sending them to "Limbo".

Now, after years of preaching this rubbish, they have decided that they have had it wrong all these years, like so much of the rest of the crap they teach!

The human race does not need this nonsense any longer!

Let’s say you don’t get your kids one of those childhood vaccines. What would happen? They could contract a deadly disease and die, is what could happen. It would be foolish to think the spiritual realm is different. Baptism is a spiritual “vaccine” against Original Sin