Most of the people who respond here are NOT attorneys and even among the attorneys who respond, not everyone will be licensed in your state. Therefore, we can provide suggestions, but only an attorney can provide legal advice.
Courts act in response to "complaints", which might be called motions or pettiions. Obviously, we can't read your document, but I think that the odds are over 98% that your ex did file with the court and asked it to find you in contempt of court for failure to pay court ordered child support. If your ex filed for state assistance, it is quite possible that the child support enforcement agency required her to file or filed on her behalf. Just because you are unemployed does not mean that your kids don't need food, clothing and a place to live.
You will need to defend. You also may want to file for a modification of your support obligation on the basis of a change in circumstances. Please note that any modification request, if granted, will NOT be retroactive in nature.
You will have to contact your local legal aid society to determine whether you qualify for services.
To my knowledge, Unemployment Insurance (UI) benefits can be levied for child support and child support arrears.
You owe whatever your child support order says you must pay until such time as the order is changed. So, yes you likely do owe back child support. In many states, arrears also accumulte interest.
I suggest that you re-examine your list of reasonable and necessary living expenses and determine if there are options for you to trim those expenses so that you can meet at least some of your ongoing child support obligation. I also suggest that you be prepared to show the court that you are making a good faith effort to find a new job.
In future, do not regard court orders as suggestions that can be unilaterally disregarded because of a change in circumstances. You really needed to go back to court as soon as you lost your job and realized that you would have a problem making these payments. You appear to have waited at least four months. Not a very good fact pattern in court.