I'm new to this so go easy

The survey on my new house reported damp in the concrete kitchen & bathroom floor and suggested digging up, laying damp proof membrane then re-laying concrete.
Rather than re-lay concrete, I'm going to put a wood floor down to make these rooms consistent with the rest of the house and allow me to run the plumbing for these two rooms under the floor (rather than around the skirting as is the case currently).
I've had a little go at the concrete already and it looks like a fairly thin layer of self-levelling compound then about 3 inches of concrete on top of rubble (it came apart with a few hits of a club hammer).
Assuming I use 2"x6" joists, what is the minimum depth I need to dig the floor to? Once the floor is up do I still need to lay a damp proof membrane before I install the wooden floor?
I'm planning on putting a few air bricks in for ventilation in case that makes a difference.
Thanks for any help in advance.