For one thing none of that should have ever passed a foundation inspection. None of it comes even close to meeting any building codes I've ever heard of.
If I had of caught him doing that stuff on one of my jobs I'd do my best to make sure it never happened to anyone agin and sent him down the road.
Without a footing below frost level and 24" wide that slab is going to just crack and sink, throwing off the whole building.
Digging under the slab now and trying to set rebar then back fill with concrete is going to be a very expencive and back breaking job, and may or may not work because new concrete does not like to stick to old and there's no good way to force it up tight to the bottom of the old slab, plus what's going to hold the building up while you do it? Your going to have to build temperary walls inside to hold it up.
I'd call some companys that do what's called mud jacking and see what they say about it.
As far as under the coloums now your stuck with cutting out a 24 X 24" hole in the slab and digging it out at least 8", setting rows of rebar on chairs and poring concrete rated for 5000 Lb..