I am in the process right now and have done several moves.
You are correct in saying they won't move chemicals (if they know about them) or any perishable items including liquids.
As for lumber, yes they will move it. Each individual contract carrier will charge their own price per pound for what you go over. E6 with dependents is somewhere like 10K pounds that will be paid for my the government. Check out move.mil for more information.