Thanks to you and your family for taking the time to play the game!
As for the questions:
Q:
Do you have to take all moves when you're attacking?
Do you have to take all the moves when you're trying to land on a planet?
A:
No.
Q: Can you trade cards?
A: I don't see why not, that could be interesting.
Q: Do you have to reveal your hand?
A: No
Q:
Can you play multiple production cards on one planet?
Can you play multiple production cards on one turn?
A:
Yes
Q: What happens when all of your pieces leave a planet?
A: That planet is no longer occupied and you will not be able to put created units on that planet. If you don't occupy any planets you won't be able to produce new units.
Q: Does the "Tiny World" card place a planet randomly?
A: Not sure if you mean the "Planet Building" card or not. The "Tiny World" card does not place any planets. The "Planet Building" card places a planet randomly.
Q: When generating the starting four planets, do you have to place one in each 4x4 quadrant, or does the dice roll determine that?
A: You have to place one per quadrant. This is to avoid too much of an advantage for one player as they could all be placed in one quadrant. Probably should have made this more clear.
Q: Do you have to own the destination planet to teleport to when using the "Matter Transport" card?
A: Yes that was the intention.
Q: When a person dies, does the person that killed him get the cards/planets?
A: I didn't think of that. I only ever playtested with two players so I guess I never stumbled upon this. I'd say it could be fun to try to give the cards to the other player. Planets however would not be transferred as if a player has no units he does not occupy any planets.
Q: Can you play more than one "Super Duper Lazer Beam" in an attack/defending?
A: Yes
Q: Can you move on top of a player and attack from there?
A: No
Q: Do dead units disappear forever?
A: Well you can use the same tokens to represent new units that you produce if that is what you mean.
Q: How many units are you allowed to have on a planet at a time?
A: I realized it was probably a good idea to have a limit after I finished the game for LD.
Q: If you are attacking a planet, do all the tokens stacked on the planet combine for defense?
A: Yes
Q: If someone attacks a piece outside of a planet, can other units adjacent to the one attacking help defend?
A: No, the intention was that it was safer to move in stacks than individual units.
Q: If multiple units can defend, do all the defending units die upon loss?
A: Yes
Q: If I understand the instructions correctly, you can only attack/invade at the beginning of your turn. This means that you have to move next to someone, wait for them to try to attack you, then you can retaliate when it's your turn again. Is that right?
Yes, post-LD I actually changed quite a lot of the rules which made it a quite different game. One of them was that the planet actually added a "unit" for the defending player but only if he was defending which discouraged the defending player to attack as he is more powerful when defending.
Thanks again for trying the game out! I've taken most of your questions and improved the original ruleset a bit to hopefully make things more clear and improve the rules a bit. You can find the updated ruleset on the Google Docs version.