1. You can give it a shot (I guess it will make you feel like you're doing something). The truth is both the live-chat and phone agents see exactly the same thing. If your case hasn't been entered into the system by the Chicago Lockbox folks, neither of those agent will see anything regarding your application.
2. Maybe your file got assigned to someone who subsequently went on vacation or got some emergency preventing them from working and your case file wasn't subsequently re-assigned? Maybe the agent that signed for the form, upon delivery, took a while to forward the package to the central processing section? Yes, case files something get "stuck" in the system for some reason which we're not aware of (happens now and then), it is unfortunate when it happens to you. You can re-send another package at this point, if you like. Be prepared for the possibility of both packages been accepted and put into the system, should you decide to send in another package without the first one being rejected (your fee once cashed is non-refundable). Of course you can attempt to stop payment on the 2nd package if you get the acceptance notification about the 1st after you've sent in the 2nd package (and before it gets processed).
3. Legal consequences for lost packages by the government?
Maybe you should go talk to a lawyer.
4. You can't assume no-one gets TB in the US. Anyway the decision to re-sign the medical form with or without a new exam depends on the CS involved. They make the call on what they are willing or unwilling to do.