Bend the rules, don't break them
The first thing to note was the whole no-astronauts-and-no-spaceships bit, ... everyone stopped going into space.
Even assuming that we could accept that (we absolutely can’t),
Except we can accept this. This issue has been written about considerably. The motivation for going into space was the Cold War. No Cold War, no need to go into space. The prospect of humans ever seriously travelling into space has been in doubt for a while. Not sure what the missions and continued talk about Mars (or the Moon if you're Russia) has done to that position.
Unfortunately you're right about everything else that was wrong with the episode.
Great Sci Fi bends the rules of science, it doesn't break them.
Time travel - no problem
Unusual alien physiology (regeneration) - alright
Boxes bigger on the inside - that might work
("basic, secondary school level physics" has nothing to say on these matters, so yes they are allowed)
magically appearing and disappearing mass - NOPE.
Being on the moon and seeing Earth in darkness (as they turn off all the lights), and then travelling very quickly to the Earth, seeing the moon hatch, and it being day time! - NO NO NO NO.
This is writer Peter Harness' first Doctor Who episode. I'm not keen on there being a second.