Your channels should be 1,6,11 at a minimum. They will not overlap. Two channel separation is too small. Spanning tree is a protocol that is more switch oriented than WAP oriented. I think your problem is 2-fold…. channels too close and also the building configuration. You should hire someone to do a good wireless spectrum survey and give you an idea about adequate coverage for the facility. This is a specialized area and is not one for just anyone to work on “by the seat of their pants”. It would be money well spent and ensure the systems/services perform as desired.
You should definitely have switches on each floor unless you can ensure that the cable length does not exceed 100 meters between the client and the upstream switch.