1, 2. Read this:
quora.com/Who-would-win-in-a-war-between-Russia-and-the-US/… It appears very well sourced. TLDR: Nuclear winter is a myth, there are not nearly enough nukes to kill even a billion people, and radiation quickly becomes nonhazardous. The danger of nuclear weapons is constantly overstated and exaggerated.
However, even if there was a nuclear winter, living on earth would still be easier than living in space. Even gamma radiation will be stopped by a few meters of dirt and concrete, so bunkers are definitely viable. There would still be at least aquifer water, desalination (energy expensive, but cheaper than space travel), oxygen, etc easily available. Also, how would nuclear winter stop energy generation? Solar might become nonviable, but you could still use fossil fuels, geothermal, fission, or fusion (which you probably have if you have interstellar spacecraft).
3. Why would you not have the technology to move asteroids? If you can move a ship the size of New York, you could attach the ship to an asteroid and move it towards Mars. Also, there probably isn't an invading alien fleet, or else invading Mars would be pointless, as they would just come and kill you anyways. Also, if an ancient civilization is trying to kill you, you are certainly doomed, and running away will not change that. There is no stealth in space (look it up), so no matter if you flee or not, they will know where you went.
Continued: If you flee into space, they will simply chase you down with a larger fleet or with a massive volley of intelligent RKKVs and kill you. If you stay where you are, they will melt your planet with a Nicoll-Dyson beam or slam a few billion RKKVs each with the energy of the Chicxulub asteroid into your planet. If you try to expand into your solar system, they will melt every planet in your solar system or hit every one with an RKKV, then follow it up with a massive invasion force of millions to billions of self replicating war machines.
Interstellar civilizations are incomprehensibly vast. Even a Kardashev 1.something civilization with a partial dyson swarm would outnumber us billions to one and would likely have an economy hundreds of billions or trillions of times larger than ours, as well as better technology.
They could exterminate us with the same effort we use to go to the supermarket. In fact, if there was a homicidal civilization near us, we'd already be dead. It is possible to detect atmospheric compositions of exoplanets with telescopes, and they would presumably go about melting every planet with an atmosphere capable of supporting life that they detect with a Nicoll-Dyson beam before anything more advanced than a bacterium emerges.
See Isaac Arthur's Interstellar Warfare and the first few minutes of his Space Warfare videos for a sense of the scale of these civilizations.
6. It does not matter if they also use carbohydrates and fats and proteins. Their versions will probably be completely different. Remember, even if their basic sugar monomer is C6H12O6 like ours, it will probably have slight structural differences that make it indigestible to us. We need specific enzymes to catalyze the breakdown of every type of molecule, and we will not have the ones needed to break down their molecules.
Even if they use the exact same monomers of glucose, fructose, and galactose, their polysaccharides will probably have those linked together in different ways, and we will similarly be incapable of digesting them. The same goes for proteins and fats.
7. A proton moving at 0.2c has something like 10^-9 J of energy. Our current nuclear reactors release neutrons at 0.1c, and those don't cause huge explosions. No proton would destroy a ship. Also, a 1kg rock would only have about 100 kilotons of explosive force if moving at 0.1c. That is somewhat impressive, but would not be fatal to a ship the size of New York State.
Remember, modern spacecraft have no armor, as they do not travel at relativistic speeds, as lifting things into orbit is expensive, and and as modern engines are inefficient. With fusion, antimatter, or black hole engines, moving much more mass will be possible. Also, larger spacecraft will be able to have much thicker armor, due to the square cube law (doubling dimensions gives you eight times the volume and only four times more area, so thicker armor will be proportionally lighter).
Because of this, I would expect such a large spacecraft to have very thick armor, with Whipple shields to fragment incoming projectiles on top. It would probably also have multiple layers of internal armor, so even penetrating objects would not go through all the way. The Castle Bravo nuclear test of 15 megatons, which released more energy than this 1kg rock, only made a 250 ft deep hole, and that was in soil and rock, not in ultratough futuristic spacecraft hullmetal.
Any large asteroid will be detected easily by the ship's sensors at massive ranges, while any small debris, though harder to detect, will be easier to vaporize at close ranges with either lasers or nuclear missiles, and won't cause fatal damage to the ship.