1. What would happen if you dug down, would you reach the other side? You must at some point because that would break the laws of physics by there being infinite matter.
2.
Gravity dose not exist, the earth simply accelerates up at 9.8 meters per second BUT
It would reach the speed of light, in which time stops.
1. That's irrelevant in practical terms. In reality we have dug down only a few kilometers. That's nothing. FE could be 100 km or 1000 km thick and it would be all the same to us.
2. FE reaching lightspeed is not a problem in itself.
The first big problem is that the starlight would be blueshifted because we would be moving towards the stars extremely fast. The stars shining vertically down on FE would have a larger blueshift than the stars shining on FE from the side.
The second big problem would be that FE would ram into the interstellar/interplanetary dust. That dust impacting on any matter (be it an atmosphere or a dome) causes massive particle-radiation.
https://en.wikipedia.org/wiki/Air_shower_(physics)
If Earth were flat, those particles would come into the detectors at a different angle than what is observed. For example, scientists have detected stellar neutrinos coming from underground. They are byproducts of the nuclear reaction in the sun and they passed through the whole of Earth before reaching the detector. And we can tell that they must be stellar neutrinos (as e.g. opposed to neutrinos from man-made nuclear reactors on Earth) because they have a specific lepton-number ratio that cannot be produced by any natural or artificial neutrino-source on Earth.