Can you take your child out of the country without your spouse's
permission if you are happily married and there are no legal or
custodial issues?  What about if you are divorced and you have joint
custody of the children? Do you need your ex-spouse's legal permission
if you are divorced and want to take your child out of the country?

This would depend on the country you're leaving as well as the country
you're entering. In any case, I'd believe that $2 is not a sum that
would encourage Researchers to have a serious look at your question.

In the US, almost certainly yes.  If you are divorced, there's a good
chance no.  However, the question implies underlying problems and
nobody is going to be able to give you a definitive answer, including
me.  You need a lawyer for that.

Not enough information.  For example, if one goes by airplane to
Mexico with a minor child, one needs a signed permission by the other
parent -- if they think to check...

I once got a paper from the Mexican Consulate in McAllen, and my
answer was written on the back of a form which allowed a person to
sign that they were the true custodial parent and had the legal right
to take that child in to Mexico.

Ask your airlines or travel agent.