First, check with your local AHJ about siting, setbacks, and if there are any easements where you intend to do work. Do you have a sealed boundary survey showing your plot corners? And easements, retention areas, drainage requirements, or other encumbrances to where you wish to work? What size building are zoning ordinances going to allow/permit? Are there restrictions on the appearance, does it need to match your dwelling architecturally? Do the fenestrations, siding, and roof covering need to compliment your existing dwelling?
You're better-off I believe, to demo everything there, including the slab, and start anew. If you want to try salvaging what lumber is reuseable, or that could be sold, you could try that.
If the floor now is lower than the surrounding grade, you need to raise the building footprint above the yard grade. If this means that runoff water might run onto your neighbor's plot, you may need to allow for a retention area on your plot, until the water percolates into the ground. You may be required to submit a soil test core to see how much weight in new construction the existing site can bear.
Your trees or your neighbor's trees are going to continue to be a problem. The root structure is often the extent of the tree's canopy. 'Clearing and grubbing' is removal of plants, their roots, and that is what you need to do there. Forget attempting to do it with a hand-tool approach. You need a front-end loader/backhoe to do it properly, and then lifts of compacted material to provide a stable base for the footprint. Non-vegetative! Do not expect to mix the ripped-out roots back into the building footprint as 'fill.' That material needs to leave.
You will be much happier doing it properly, as it will be far-less problematic down the line. Reading the land development code for your jurisdiction is the first thing to do, looking at what is allowed in your zoning district. Then go speak with whoever you need to in your community who has oversight of building projects. It's possible that more-restrictive regulations for things like % of plot coverage and setbacks for new development have changed since that structure was built.
