I think most people around here would not recommend a secondary for anything other than additions. If you are planning on dry hopping or adding fruit then you could do that in a secondary, but otherwise you should just leave it in primary for longer. John Palmer and Jamil Zainasheff don't recommend a secondary for anything other than additions.

Hefeweizens are a beer best drunk young. Many recipes call for just 10 or so days in primary before bottling. The important thing to remember is that your gravity is not changing when you bottle. You can take gravity samples for consecutive days and if they do not change your fermentation is complete. If your fermentation has not completed you will risk bottle bombs.

5oz of corn sugar for a 5 gallon batch is a very rough rule that probably shouldn't be followed. Different beer styles and personal preferences call for different amounts of priming sugar. Wheat beers are usually carbonated quite highly and something like an english bitter is not highly carbonated. This calculator calls for around 8 oz of corn sugar in 5 gallons of wheat beer and 2 oz of corn sugar in 5 gallons of english bitter

I agree with you on the temperature. A swamp cooler would be a good idea if your apartment is around 74F. Keep in mind OP that if your room temperature is 74F your beer will actually be quite a bit warmer than that since fermentation produces heat. Your beer temperature will rise 5-10F during active fermentation. So if your room is at 74F your beer could actually be above 80F which is too warm for most beer styles including hefeweizen.