, we will lose this atmosphere either way, so this is absolutely necessary for our long term success. And it will take some time.
- Surviving the conditional extremes
We need something to survive the extreme temperatures coming from below the thrusters, within the underground system (via the planet's mantle, core, etc) and from without and above (i.e the cold gas of a nebula, the heat from a near by star).
We need something to survive the extreme pressure as we build structures deeper into the planet, and as we travel closer or further away from massive objects.
Can we address all of these problems and the concerns raised by @D. Elliot Lamb and some of others in the second link? Turns out we can! We have our unobtainium: Aerogel composites!
Sufficient research could yield a host of aerogels with the necessary properties for most of our engineering needs:
Today's aerogel is already capable of maintaining functionality under extreme temperature differentials.
Aerogel's porosity gives it the mechanical properties necessary to bear high loads. Creating a composite with the right formula could allow our engineers to create a sort of planetary spring/sponge. This will be particularly useful for absorbing the forces created during acceleration and from tidal forces from other large planets.
As a bonus we can also store our atmosphere within the aerogel, killing two birds with one stone. Closer to the bore hole walls, the aerogel structures can absorb some of the steam produced from continuous wall cooling, using the aerogel as a thermoelectric catalyst to produce hydrogen and oxygen necessary for our chemistry and survival respectively (producing ammonia for food, breathing).
And while we're at it, let's grow things micro-organisms our aerogel pores :)
Eventually our scientists will figure out how to evolve a collection of organisms which can regrow what will become our symbiotic planetary host.
2a. The serious issue of building large structures
Even after removing the atmosphere, we still must support the pressure of our entire system as we move closer to the planet's outer core. Using Earth as an example, that's a 2,890 km (1,800 mi) overhead we have to deal with.
The composition of the mantle is locally a solid, but essentially a fluid over time. And its temperature (using Earth as an example) can range from between 500 to 900 °C (932 to 1,652 °F) at the upper boundary with the crust; to over 4,000 °C (7,230 °F) at the boundary with the core. [shameless copy pasted from Wiki]
Can we provide an anchoring solution strong enough to withstand the heat and pressure of geological time?
I think so, in the present moment we have (predicted) Hafnium Carbide superalloys with of melting points of about 7,460 degrees Fahrenheit:
And Hafnium superalloys are already known to be excellent an neutron moderators, to shield us from the neutrons produced from nuclear reactions in the planet.
Scientists are already pursuing Hafnium Carbide aerogels!
Having strong, porous materials along with an abundance of atmospheric gases can allow us to making floating platforms which withstand the large pressures from above. Technology like this also already exists:
In conclusion, by simultaneously constructing a planetary spring/storage system, a geothermal/nuclear energy system, and all the other planetary scale systems you'd need included, you could satisfy all of the requirements of creating a traveling planet which could sustain its own needs and survive the internal and external forces of traveling intergalactic space :)