Does anyone know of a Qualified Demolition contractor who can help with permitting, Demolition and Framing?

It would make sense for them to frame the house, they are already there and they have to support the structure as they bring it down.

Maybe they can even consult with the client on what to build, and how to build it cost effectivly and after that they can  facilitate with the permits. That sounds like it would make sense as far as the construction schedule goes and it might even bring down your labor cost on the frame.

Does anyone have a company like that?

I know of one, message me for details

Demolition is a bit of a specialty.  I wouldn't be concerned with trying to find one contractor to do the demo, and the construction.

Separate the trades, demo contractors are faster with demolition work and framing contractor is faster framing them. You will need knowledge on how things will get done, before you invite bids, make sure you make a scope so you will receive more apples to apples bid. If you can't do it, ask an architect, project manager, or engineer who will draw the plans, or simply just hire a GC to do it for you. Not easy to support a house, especially if it is a tear down, has bearing walls, or need to jack up the second floor to demo the first. And you are limiting yourself the opportunity of savings and quality work by looking for someone who can do two trades in one company, if there are 30 demo contractors and 59 framing, you might only have 5 companies who can do both trades, and most probably they are a self-performing GC and have a GC license.

usually my carpenter pulls the roof off and any internal framing

The demo crew strips everything else.  

Basically I was trying to say I started Apex Demolition and frame

We- advise, pull permits, in house architect, in house engineer. We can advise you on different options to make the house "pop" or more cost saving methods.

My personal flip closed today, we are 25% complete with the demo. We sound proofed the house to enable us to work a day and night shift. I directed my framer/demo'er to focus on demoing the bathroom as that is the future location of the stairs. Soon as he is done with demo he's going to start building the stairs. I am a true believer in Parallel process and lien construction (the elimination of waste).

My goal is to complete two 1400sqft additions in 3 months. It is actually easier than new construction when you have the right team. I am a builder by trade with 10 years in the trenches and there are a ton of options you have in scheduling your project to cut cost and to speed up the schedule.

Jared