Why is the floor high in the center? - Bad workmanship or expansive soil or what?
Most likely cause of raising in the center of the slab is lowering around perimeter of basement slab. The settling at the perimeter of the slab causes the middle to lift. This is caused by settling of the subgrade from lack of compaction on the soil below. As the house was built in the 50's the amount of compactive effort applied was probably minimal at best. Expansive soils are not the norm but are occassionally the problem. There are some clays that are expansive along with some man-made material that were unknowingly used as fill in the past. If you are the only person in your neighborhood experiencing these problems it is most likely due to settling around the perimeter. If all of your neighbors are experiencing a similar situation then you may want to call an engineer.
If you remove the slab it is a good idea to make sure that the sub-base is adequately compacted prior to the placement of the new slab. Also pumping the concrete in would be the way to go. Hand or machine mixing that much concrete would be one heck of a day and the cost may be cheaper than you think to have someone do it.
As a sidenote, if headroom is a problem and you have a relatively small basement you could possibly lower the slab to gain additional headroom. As long as the slab stays completely above the bottom of the footings and water is not an issue you should be OK. Ths may gain you as much as 2 feet. Also if you would like to go deeper you could speak to an engineer who could design an underpinning system which will allow you to go as deep as you want. All of this would come at a cost
but is always an option.
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