Frank,
At this point I would not worry. While the insurance company says they will do nothing for the sinkhole in the back yard, they are at least doing testing. When the results form the testing comes back they may find unstable conditions closer to the property. While this does not sound good, it is essential for the insurance company in order to start repairs.
I would wait to do anything at this point like trying to fill the sinkhole. The last person that I talked to had two truck loads of fill brought in to put in the hole and it all just disappeared. Also putting that kind of weight on an open sink without any testing can cause it to grow.
In the mean time take pictures of how it looks now, reinspect your property for ANY kind of cracks (ceiling,floor tile,wall joint,exterior block) and take your time with it. It might be a good idea to mark the outer rim of the sink with spray paint (try orange or yellow cause it really shows up in photos) in order to see of it is growing.
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