We also have a newborn and would like to remain in our current home with her at least for the next few years.I'm willing to get creative, willing to get my hands dirty (as in demo/construction, not burying bodies...). 

My wife and daughter found a dehydrated rat in a kitchen wall during a demo.

So the details: My agent helped me find this property off-market, the previous owner had just passed and left the property to her daughter and she wanted to just get out from it and move on; its a little bit of a hoarder house (the before pics was taken halfway thru demo so doesn't show how much crap was there!)

Ok maybe.You will still have to pay to demo the damaged property and that can run from $15k to $40k or more depending on where you live so you may not have much left of your initial investment to buy that next property.And if you financed it - the bank is going to take the majority of the insurance settlement to settle the loan and you will own free and clear a fire damaged house that the city will eventually start making demands for you to repair or demolish.This is why the majority of lenders require you to carry full replacement cost coverage.

Light rehab,, demo , paint,  May need some water damage repair in bathroom. 

The funny part is we demoed all the ceilings but the slab leak didn't develop until they were put back together.

We purchased a duplex in a beach community that needed rehabbing(one half had been demoed to studs) after purchasing and applying for permits we find that fema limits the rehab to 50% of the apparaised value of the house.

I've seen rowhouse demos where you could see the interior lath of the neighbor's plaster walls from the street! 

You may have had discussions about living there, but I bet you didn't have discussions about air compressors, circular saws, and drywall demo. 

I managed the projects, so coordinating the demo and install timing on things was key and each contractor was accommodating and fairly priced.