David Wilson Homes should be looking into this not fobbing you off!
Just because it should have been built to the Robust details, doesn't mean it actually was.
Can I ask if your home is timber frame? These can be especially noisy.
It does appear to me that the staircase has been fixed to the party-wall which is why it transfers sound.
What needs to be done is an acoustic sound test. To do this, access will be required to your neighbour's property on the day. I am sure they would agree.
Write to David Wilson Homes and ask them to arrange an acoustic sound test. If they refuse, write to the NHBC and make a claim under your warranty. Make sure the acoustic engineer understands that he must send you a copy of his report and results.
One thing for sure, you should not be able to hear those sounds in a brand new home built in the 21st century! Once you have the results, you will have proof that the construction does not meet building regulations for sound transmittance and expensive and extensive remedial work may be necessary.
Finally I feel sorry for their dog! Not much of a life is it?