First, I hope your issue has been resolved, as it has been over 9 months since you posted your question. In many automobile accidents, the property damage claim is the most frustrating part of the case. Typically, the insurance company's goal is to get the car fixed for the lowest price possible. Sometimes, supplementary repairs are necessary, and you are left without clear direction about who is responsible for these extra repairs. Often, the car has hidden damages that make it a total...
As for North Carolina cases, generally you can negotiate with the adjuster who has offered money in a case like this. In North Carolina, we have a Tort Claims Act that governs some school bus cases, so you will need to know if your state has that also. The Tort Claims Act here has very specific steps you must take to achieve a successful outcome. It sounds like you could use a consult with an Illinois lawyer to learn your legal rights there.
I agree with the other poster about forming a group to complain together. There must be a source for these bugs, and it might be one of the tenant's apartments. It may require finding the source before you're going to get any relief.
It might be a good idea to get a GC to come in first, look it over, and if he/she thinks it's going to pass, ask them to talk with the inspector. Often those folks have good relationships and can get things passed through quickly. It you're not a licensed contractor, you might miss something that the inspector is concerned about.
In most states, there is a process for adverse possession. It will vary by jurisdiction, sometimes taking as long as 21 years of open and notorious occupancy. Generally, if the owner of the land gives some type of approval to the squatter, it resets the time clock. If you're interested, contact a property law teacher at a local law school and they'll tell you all about it for your state.