Answers to various comments/questions on this sub-thread:
Time dilation at 1/3 c is 5.7%, quite a noticeable amount, but not remotely close to to turning billions of years into millions.
Tidal effects are small for super-massive galactic black holes. I doubt tidal disruption of Earth-like (i.e. fairly close) orbits would occur, especially for cool M-type stars (the most common kind).
While individual particles of cosmic dust hitting the planet at 1/3 c won't be a problem, (they will simply explode high in the upper atmosphere), the energy flux hitting the atmosphere from interstellar gas would be considerable. Average interstellar space has something like 1,000,000 hydrogen atoms per cubic meter. At 100,000 km/sec every second there would be 100,000*1,000*1,000,000= 10^14 hydrogen atoms hitting each square meter of atmosphere. The kinetic energy of those atoms would be about 1000 J, so roughly 1000 watts/m^2 of heating from interstellar hydrogen. Earth gets 1400 watts/M^2 of heat from the Sun, so it would roughly double the heating of an Earth-like world until it cleared the galaxy plane. If it ran into a denser patch (all of the region in the galactic center would be denser than the average I quoted) then the heating could be 10, 100, even 1000 times higher for a bit. I think this would cook any existing Earth-like planet.
Once in interstellar space though the heat load would drop by a factor of 10,000 to 100,000 of the average interstellar value and would cease to be significant. From there on the planet and star system would evolve on their own, and a new biosphere could come into existence.