Both angular momentum of the moon-earth system and the angular momentum of the sun-all-planets system are conserved. Linear momentum must be conserved also. Finally, we must account for the changes in the magnitude of gravitational attraction.
Since the mass of earth is the same as before, the gravitational attraction from the earth to both the moon and the sun are the same. Nothing about reforming the earth inevitably changes the basic orbit with respect to both the sun and moon. It could still happen if humanity wasn't careful.
Different types of angular momentum are linked. If what you did stopped the earth from rotating, you would affect the other angular momentums.
In the sun-all-planets system, angular momentum would need computer modeling. The earth's rotation is a bit player though. Venus has almost stopped rotating and it is still there in a mostly circular orbit
However, the angular momentum of the earth-moon system would be affected if the earth stopped rotating. The angular momentum of the earth moon system ("L(total)") Lrot is the angular momentum of the earth's rotation. Lmo is the angular momentum of the moon's rotation. L(total)= Lrot + Lmo.
According to zipcon.net, Lmo is four times Lrot. If the moon got all of the angular momentum from a spinning earth, it would just orbit further away.
It is also possible such an extreme process might disrupt the earth's rotation around the sun by making the rotation more elliptical. This would make seasons more extreme. Angular momentum would be conserved because the earth would move faster when closer to the sun and slow down when further away. This is Kepler's second law of planetary motion. Earth would never "drift into" the sun.
Let's face it. More extreme seasons would be the least of the problems of the people still living there. If the earth is no longer spherical, the earth's core would cool. There would be no magnetic field protection from solar wind. We'd probably either lose the ocean and atmosphere or they would be spread too thin to be of any use.
EDIT: 08/23/2017 Your question keeps changing.
When I read the earth was being “reused,” I was picturing building a non-spherical shape using the material in the earth. Since the surface area versus volume would be greater for any other shape other than a sphere, the atmosphere and water in the oceans would end up spread out very thin just by geometry. I am guessing this was not your intent.
I also implicitly presumed that we had unknown technology that could cool the inside of the planet, letting it get ripped apart in the first place. However, the level of detail you are asking for seems to indicate you want the most realistic answer possible. The ultra realistic answer is as follows:
The earth is more like a deep fried cheese curd than a cold solid mass of rock. The Kola Superdeep Borehole in Russia got to between seven and eight miles deep before being abandoned due to temperature. See also: https://www.livescience.com/6959-hole-drilled-bottom-earth-crust-breakthrough-mantle-looms.html
Beneath the hard and thin crust, earth’s mantle is pliable molten rock and that every dense astronomical body above a certain size of large asteroid wants to be a sphere because of gravity. The force exerted on the crust at the earth’s equator creates a bulge around the equator. However, this bulge is 28 miles, which is tiny in comparison to the diameter of the earth. https://www.space.com/17638-how-big-is-earth.html
So, you would have to pull with a force much greater than the forces involved in earth’s rotation to have a shot at pulling out a significant piece of mantle. Obviously, that would take an immeasurable amount of energy. With that level of energy being inserted into the earth, it would be anyone’s guess as to what orbit or rotation would result or whether the moon would get flung out of orbit as a complication of that process. I do know that the heat lost due to that much friction would heat up the earth to a molten state.
What you could do is dredge the crust off of the earth. However, even if you took the entire crust off of the earth, it still wouldn’t be a “large” chunk of the planet’s overall mass. You’d only be able to strip the crust off earth once. The top of the mantle would take an extremely long time to cool. Earth’s crust changes really slowly. Parts of Scandinavia are still rebounding after the weight of glaciers was removed 19,000 years ago. http://www.antarcticglaciers.org/glaciers-and-climate/sea-level-rise-2/recovering-from-an-ice-age/ Likewise, earth would be a ball of molten mantle for the foreseeable future after you stripped off the crust.