Cook County Sherriff came yesterday to my apartment building to notify us that this property is in danger of foreclosure. As a tenant in the situation, what are my rights? Do I have to continue to pay rent, while this is happening? What am I supposed to do?

If you live there, legally you are suppose to be paying rent.
But you need to be looking for someplace to stay, there is little notice nromally after foreclosure to move

NOTIFY how? Was there an actual NOTICE in writing or just a deputy telling people something was going to happen?

Talk to your landlord and relate what happened and ask what is going on.