1. If Apollo was faked, what motive would NASA have to fake the near-catastrophe that was Apollo 13? A publicity stunt for funding perhaps?
Have you seen the movie? It's just a hell of a story. Also, it's possible they needed a reason to stop.
2. Moon rocks can be seen (and sometimes touched) in many museums. They don't feel like rocks on earth... Where do they come from?
Volcanic glass doesn't feel like other rocks on earth. This does not imply that it comes from space.
3. So if all NASA does is spread propaganda and "embezzle funds" (as per the wiki), then who invented all of this space tech?
I'll be honest, I'm not convinced of the grand NASA conspiracy... but inventions like LEDs also don't require going to space.
4. What happened to Amelia Earhart?
Your guess is as good as mine. Does the shape of the earth make a difference?
5. A famous philosopher once said that the simplest explanation is usually the most correct. What caused flat-earthers to doubt the model they were taught in childhood?
I guess this is a difference in our interpretation of what 'simple' means, but the moving, spherical earth solar system model is far from the simplest explanation. Even a geocentric system is simpler. Go outside; look at the horizon. The simplest explanation is that the earth is stationary and flat. Is it correct? I'm not sure, but I'm starting there.