I'm not completely sure... But I think that each stadium for either league is built based on whoever owns the team of said stadium and the current place that the team is living in (Ex. The Atlanta Falcons probably have to work with the city of Atlanta since that is where the dome they play in is located, and the training camp is located in Flowery Branch, so the owner of the team probably had to work with Flowery Branch on where the camp would be and all the other aspects of building it there.)
Basically, I think that, unless each league has certain regulations on size of the stadium and the such, that just pulling a random team out of the American and National Leagues and seeing whose is bigger would depend on how the owners and the city made it. If a random National league team stadium can hold 80,000 people in one game and a random American league team can hold 90,000... Then that American league team's stadium is bigger. That's not to say that another National league team hasn't built a bigger one. If you do still want a definite answer, you might want to try looking up "Spectator capacity in American and National league football team stadiums" on Google. Try digging around a bit.
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No. Both leagues have to follow the same rules. The only difference is the National League chooses not to elect a Designated Hitter, as that rule is optional at the beginning of any game.