There's only two categories of FTL flight: those schemes that are rendered impossible by relativity, and those rendered impossible by every practical reason imaginable. All of the latter need literally astronomical amounts of energy to work, some even more than there is in the observable universe! Where do you get all that juice? How to sustain its generation? The so-called "real-life warp drive" imagined by Doc Alcubierre for fun may not violate known phyics but runs into the same problems. How do you produce all that exotic matter? How do you form it into a bubble? And, since the hypothetical ship inside this deformed local spacetime can't interact with the outisde universe, how does it manipulate the exotic matter sphere and steer or stop? Under scrutiny, FTL in all its forms turns out to be a pipe dream.
Fallow, you majoring/minoring in physics? Then be so kind to get back at this; a few years back I heard on a pop-science show even spaceflight at relativistic speeds is, uh, non-trivial, as you're not only cooked by star light turned into high-energy UV radiation by the Doppler effect (or was it Lorentz contraction..?) but also that your ship's mass increases the closer you get to lightspeed and it collapses into itself once you cross the threshold.
Geology. Almost all the physics I've studied has been classical, so I've spoken with my brother about this subject, who studies astrophysics.
You're correct that objects traveling at great speed become more massive
from the perspective of a bystander in an inertial reference frame.
Inside the spaceship's point of view, however it would appear as if the ship's mass was constant and that the rest of the objects flying by were becoming more massive as you accelerated relative to them.
My brother reasoned that, because from the perspective of the ship, its own mass isn't increasing, that it would not collapse upon itself under its own gravity.
I am not sure about the speculation about the Doppler effect; how fast would you need to go to make the frequency of incoming light high enough to cause damage, and would their concentration ever be high enough to be significant? I don't know.
edit: assuming that space is full of microwave background radiation 3kelvin, and wavelength 0.03m, traveling at 99% light speed would only cause the light to appear to have a wavelength of 42mm, which is still in the microwave region.
[gamma= 1/root(1-(vv/cc))] = 1/root(0.99*0.99/1) ~ lorenz contraction by a factor of 7]
Visible light would shift to UV, and UV would shift to X-Ray.
About 500 photons of the cosmic background radiation occupy each cubic centimetre.
You would need to travel at about 0.99999999995 light speed in order to shift these photons up to a frequency high enough to give you a sun tan.
There would be about 10^5 more photons in each centimetre cubed, because space would contract relative to your ship.
And since you're traveling at light speed 3*10^10 of those parcels hit each square centimetre of your ship each second.
Hence the flux of UV would be about 3*10^10 * 10^5 per square centimetre
UV has about 4eV of energy per photon, or 4*1.6*10^-19J so
3*10^10 *10^5 *10^4 *4*1.6*10^-19 = 19.2watts per square meter, which is about 6 times lower than the amount you need to be sunburnt [120 watts per square meter]
Hence, if you flew through space at 0.99999999995 and there was only microwave background radiation in your way, you wouldn't even be able to get a sunburn.
So I doubt that you could cook a craft by flying it at relativistic speeds. 
Maybe we can still figure out how to make something like that using hydrogen bombs/explosions only. Current hydrogen bombs are ignited by first using regular chemical explosives to set off a dirty plutonium fission nuke, which sets off a clean hydrogen fusion nuke (first creates radioactive dust fallout, second creates helium). If we could set off a hydrogen fusion bomb directly, it would create radiation that won't last, and would be "clean."