Decided to make a list since the same questions keep resurfacing. A lot of these answers were given in the film, some are common sense, or would be known to anyone familiar with Superman, but if you aren't familiar with Superman or missed something I hope this helps.
1. How did the suit get on the 20,000 year-old ship? Did Jor-El's hologram print it out or something? Makes no sense, man!
No, Jor-El's hologram didn't download it. It is a 20,000 year old suit. The suit belonged to the explorers who were aboard the scout ship. The reason it has an 'S' already on it was explained in the film. The symbol represents 'HOPE" not the El Family, aslo if you note during the scene where Holo-Jor-El projects the history of Krypton for Kal-El, you will see that the space colonists were planting flags with the 'S' on it at the planets they arrived at. Not to claim for the El family, but to symbolize the hope for Krypton and it's expansion. The symbol is not used in the same way as previous films or comics.
2. Why did Zod take Lois in the ship, he aready had Kal-El? Stupid!
Zod took her for the same reason the Feds did.
Zod wanted to extract every bit of info on Kal El's secret identity, and if you recall, the national news networks had interviewed a blogger who stated that Lois knows who the alien really is. Zod monitored the news, apparently.
When she meets up with Clark after he passes out, she explains that they went into her mind and learned everything she knew about him. Zod didn't know exactly what she knew, just that she knew something. As a brilliant tactician and military mind, to make sure that he had all the possible information he could, he took her to see exactly what she knew.
He didn't want to take any chances and wanted to know everything he could.
Remember, at this time he wanted the codex, not Kal El specifically.
The location of the pod Kal El was sent in could have housed it.
Anyone with any information (including Ma Kent) were questioned.
3. Why not just read Ma Kent's mind instead of choking her? DUH!
Obviously there was no need. The choke worked.
4. Why didn't the Kryptonians evacuate the planet when Jor-El told them to?! They have spaceships, just fly away!
No one believed Jor-El. One man saying the planet is going to explode isn't enough to convince the entire population to leave it. Especially if that man's theories are refuted by other scientists.
When a man tells the world that the planet is going to end very shortly (by comet/meteor, God/rapture, natural disaster, etc.) on Earth, no matter how respected, what happens? No listens, and they're always wrong.
Krypton was an isolationist society, heavily controlled, and mired in bureaucracy. They stopped exploring (Jor-El explains this) and were content to remain where they were with what they had. The leaders were arrogant and stubborn. The implosion happens so violently and suddenly there was no time to leave. None of their ships were prepared for long distance space travel with no clear destination. If someone DID managed to get through the violent quakes and volcanic eruptions and make it to a ship that wasn't toppled over how far would they get with the little food, water, and fuel that they had?
5. Why didn't Kal-El lead Zod away from Metropolis? He did that in Superman 2, so why not do it again?
The fight between Superman and the Kryptonians/Zod in metropolis during Superman 2 and Man of Steel were not for the same reasons. In Superman 2, Zod wanted to rule Earth, and defeat Kal-El based on his hatred of his father. In Man of Steel, Zod explained his goal for the planet before the fight. To kill every human he could. Illustrated when he tries to incinerate a family. The fight in Man of Steel was to keep him from doing that. If Kal-El suddenly left Metropolis hoping Zod would follow to some desert or Antarctica, he would have seen Zod stay behind to continue his destruction of the city.
Zod in Superman 2 wanted to rule humans. Zod in Man of Steel wanted them all dead. Two different objectives for the fights taking place.
6. He could have just flown straight up though and took him out of Metropolis, right? So stupid yall!
At the point where they were fighting Zod had learned to begin to master the ability to fly. If Superman tried to fly up, Zod simply had to fly down. What would happen when this occured would have been a stalemate which is pretty much what we saw. Zod could have easily resisted Superman's attempt to pull him up.
7. Since when are there Kryptonian "avatar" Dragons? So ridiculous!
Since October 1, 1952.
http://i.imgur.com/dqAauXm.jpg
Kryptonian dragons exist on Krypton in the comics and one recently appeared in the latest version of Superman comics (New 52) which he killed.
In order to have a new Krypton fully restored Zod needed two things: 1. The Codex and 2. A suitable world. He found both on Earth. Could he have left and found another world to terraform? Absolutely. He could have also simply adapted to Earth's atmosphere and lived peacefully with the humans. He didn't want that. He viewed humanity as inferior and irrelevant.
Zod was no different than the many human conquerors throughout history that went to new lands and wiped out the native population to take everything, rather than live peacefully or find another place to settle. He's a psychopath from an isolationist world.
8B Superman could have convinced him to leave, why didn't he?? Lame.
He did try. During the scene where Zod entered his mind. He begged him not to do it. He literally screamed 'NO!" Later holographic Jor-El also tried to change his mind, and Zod erased him entirely. He didn't want to change his mind.
9. Superman doesn't kill! Why did they make him do it?
Superman DOES kill if he has to. In fact, he killed Zod and Faora in the comics at one point before the retcon. This however doesn't negate the fact that he did this, as well as the fact that recently in the New 52 he planned to kill again when he had no choice:
http://i.imgur.com/PgWkrwT.jpg
He also killed Zod and the Nuclear Man in the previous films.
No. He said he let his father die because his father believed the world wasn't ready. His father refused to let Kal save him because he was protective of his son and feared what might happen to him. Not physically, but psychologically and emotionally if the world rejected and feared him. Maybe even worshiped him.
11. Why is Perry White, now Black??!
Because he's played by Laurence Fishburne.
This is consistent with the battles Superman had with powerful villains in the comic book:
http://i.imgur.com/2QIeF6g.jpg
...and the cartoon where he punches Darksied through 7 skyscrapers and into the busy streets in the middle of the city:
http://www.youtube.com/watch?v=VoJ2Bd41zsw
As well as the previous movies where he threw Zod through a building and almost killed the people in a cab:
http://www.youtube.com/watch?v=vkfUUqowQjA
I am sure he would be concerned if innocent people died, but clearly some things were unavoidable. This never bothered people before, I guess.
The powers came from the sun and the environment of the planet Earth. They could likely have retained their abilities in a new planetary atmosphere and gravity level with the same sun in some capacity or degree.
14. So Superman can now breath underwater and in space? Come on!!
No, he can't. He held his breath. Kryptonians breath like we do. That was why Zod and his crew wore helmets on Earth.
15. Why was Superman coughing on the ship? So dumb!
The ship had a Krytponian atmosphere, like how the Star Trek Enterprise has an Earth one. Kal-El's body was used to Earth's atmosphere after breathing it for 33 years and the sudden transition overwhelmed him. Lois had a helmet on to protect her. He did not.
16. Who suddenly rebuilt Metropolis??! Clark just walks into the Daily Planet and everything is suddenly normal?
Time had passed.
17. Why no use of the John Williams theme?
It's 2013. This is a new Superman and they want a new score.
18. How did Clark know about the ship in Antarctica?! Silly.
Using his super sense of hearing. He overheard the soldiers in the bar talking about what was discovered there. He somehow worked his way into being a part of the crew that did menial labor. Once he got there, he then used his X-ray vision to see exactly where the ship was. Once he found it, he used his heat-vision to bore a tunnel straight to it. That's why the tunnel Lois entered was dripping and had running water in it. He literally just made it.
19. Why did the Council send Zod to the Phantom Zone when the planet was about to be destroyed?
Jor-El was the only one freaking out about the planet's destruction, and he's killed by Zod before he and his goons are sentenced to the Phantom Zone. If he had been alive, then there would actually be a legitimate reason for this question, because you can assume that were Jor-El alive, he would object to sending Zod off planet. But he got killed, and I think the writers did this on purpose specifically so there would be nobody to object to sending Zod away.
20. Why did Superman lose his powers aboard Zod's ship ?
When Superman is restrained to the operating table (for lack of a better word) Kryptonian criminal scientist Jax-Ur explains to him that his strength derived from the Earth's sun was neutralized. The Kryptonians had the technology to enter his mind aboard their ship, it stands to reason that they also had the ability to neutralize his strength. One could assume it could be some form of technology that generated a form of Red Sun Radiation or even possibly an atmospheric configuration that they used because it was clearly planned, but once Jor-El took over the ship, it ceased to function (alarms went off) and Superman's strength returned instantly, but it isn't explained exactly how they did it. Just that they did it.
EDIT: Forgot to add that Jor-El also once downloaded into the computer, changed the atmospherics back to the Earth composition. This also benefited Kal-El. Whatever device they were using ceased to function at that point.
21. Why did Superman get weakened aboard the ship? What the?!
http://www.imdb.com/title/tt0770828/board/thread/216037205
