Can I apply for a 60-day extension of stay in Thailand for having two half-Thai children after divorce?

that is the reason that people normally apply for it (other than marriage to a thai) so you should be fine

This is a list of the general requirements, you should go to your immigration office and ask what their requirements are

It goes without saying but I will say it you have to have the reason you're getting an extension with you at the immigration office ( so at least one of your children has to be there when you apply)