if it has a concrete floor, it has probably been filled in relatively recently after the original wooden floor rotted, and is probably damp unless done, properly, in the last ten years or so. It might have an old stone floor, which will be damp. Air bricks are only found under wooden floors.
If you have a damp wall and a damp floor, it will be difficult to cure. They might be planning to cover up the damp on the inside with the slurry and plaster or drywall over it so you can't see the damp. In time it may reappear, higher up, or on one of the internal walls leading off the wet wall.
I would be reluctant to take advice from a company that makes its profits by selling damp-proofing cures. When you say they want to apply a new DPC I suspect they want to inject chemical into the wall. This is not much good. If you can find an independent surveyor who specialises in damp, and does not sell anything so has no axe to grind, you have the best chance of trustworthy advice. A local person should know the conditions, building method, and local people who can deal with it.
Walk round all sides of your house looking for an original damp course. It will probably be a thick mortar joint with a double layer of slate in it, and will be about nine inches above where ground level used to be when the house was built. It may or may not be present. Observe if at any point anything has been put against the wall above the original ground level and bridging this course.
Render should not go below this level because it provides a path for damp.
You say the front garden is built up. Is this wall damp?
You say another wall has two feet of earth against it, I presume you mean above floor level. Is this wall damp? This should be dug out. You can put a path or patio there if you want, with a retaining wall, sloped so rain runs away from the house.