Minor frame challenge
Unless they have some sort of quasi-magic FTL drive, they have to be much higher up the technology ladder than we are to be able to make an interstellar trip in something 10-20m in diameter. Now, if you're using the Road Not Taken route, that's reasonable enough, but otherwise there are enormous questions of fuel, energy, and life ...
This has no answer, under the stipulation of "The technology should be as close to our current level."
Your requirement is for a SSTO, fully reuseable orbit-to-surface-to-orbit vessel.
There is NO possible chemical fuel propulsion that can achieve this, at anything like the sort of mass fractions we can achieve. You would need to make your 11000 ...
Your concept works for the propulsion numbers.
Where you have a problem is with
Water (or liquid hydrogen, I haven't decided yet) will be pumped over a deuterium-deuterium fusion reactor to super-heat it for the purpose of thrust (explosive vaporization)."
The temperature required to reach your stated exhaust velocity are quite ...
Getting a spaceship "100% airtight" is close to impossible, anyway, so they just settled for "reasonably airtight" and carried extra air. After the disaster the speed of atmosphere loss will go up. Can they find most of the little leaks?
Solar cells won't work very well that far out. Did they have a nuclear reactor or ...
We are not always scanning the sky for threats. According to this resource, space agencies perform this task intermittently, often a few years between scans, when funding and resources are available.
When an object is detected, it's orbital parameters are determined and it is added to a catalog of Near Earth Objects. The present-day positions are ...