Who will survive nuclear apocalypse?

You might have guessed it right. Cockroaches.

Cockroaches are remarkably tolerant to radiation. A rem of 800 (A measure of radiation exposure) is deadly for humans. Cockroaches can survive upto exposure of 70,000 and above. That's a lot of radiation.

Why?

Slower cell cycles:

Cells are most sensitive to radiation when they are dividing. Some of the human cells constantly divide, thus providing more room for the radiation attack. But because of the simple bodies and slower cell cycles of cockroaches which moult about once a week at the least, cockroaches pose greater opposition to the radiation.

Now that does not mean heat cant kill it. In case of a nuclear blast cockroach has to find a distant safer place far enough from the blast radius to survive. 

Why do some jets leave white trail in the sky?

No, those trail are not the smoke from the plane engine. They are called Contrails, shortened for condensation trail. The science behind it is same as on certain days in winter when you exhale you can see your breath.

Jet fuel is actually made of carbon and hydrogen. When it burns with oxygen, CO2 is formed along with H2O. When this water mixes with the wet cold air, it condenses and ice crystals are formed. How long will these contrails last depends on how wet and cold is the surrounding air. When the conditions are pleasant, they spread across to look like Cirrus clouds.

Now, why do only few planes leave contrails?
1. Altitude - Contrails usually form at altitude above 20000 feet where the air is at -40 to -60 degree Celcius.
2. Engines - Some engines leave contrails and some don't.
3. Dry air - Contrails are not formed when the air is dry.