Last Friday (Nov 7th, 2011) I found a court Summons hanging on my post box as I walked in my apartment entrance.
It's titled "In the Circuit Court of Cook County, Illinois Domestic Relations Division" by the Illinois Department of Healthcare and Family Services, ex rel.

I am the respondent for this case and am summoned to require to file and answer in this case and appear in Room 200, 28 N. Clark St. Chicago on or before Nov 1st, 2011.

My wife left with my daughter unannounced to California and though I offered to help with finances she refused and now I am faced with this Summons. I feel this quite unfair about this and without getting into details of why I feel unfairness, I need your help in helping me navigate through this horrible predicament. I don't have much time but I need to learn, do research, know my rights, and check legitimacy/legality of this action.

Fairly simple, your wife left and is filing for court ordered support. That's her right. You need to prepare your financial information to show to the court so support can be determined.

You could, of course counter file for custody and visitation.

If a man left his partner taking his child with him and moves to Hawaii and is unemployed, would he have the right to force his partner to pay child support? It would seem awful convenient to demand money while it also appears to provide little accountability against such one-sided action.

If a man left his partner taking his child with him and moves to Hawaii ...

You first said your wife went to California, now you ask about a man taking a child to Hawaii.

Which is it? Or is the idea that a "wife" is a female spouse now obsolete?

would he have the right to force his partner to pay child support?

Force is not the right word here. Applying the law is the concept. If the non custodial parent is working and the custodial parent isn't, the law provides for support. The NCP can apply for custody. The fact that the custodial parent took off without prospects will be considered.

But you need to put aside your anger and start thinking about what's best for YOUR child. Because that's what the courts are charged with doing.

It is not if the man or the women has the custody of the child, it is WHO has them, if the man has the custody yes the wife pays support,

The issue is when she left, you could have filed in court to force her to bring the kids back, but you did nothing,

When she left, you could have filed for visitation and had child support set, but you did nothing,

So yes, it is perfectly right and fair for her to file and get child support