To answer these wonderings:
Applications typically don't open up till mid-summer. The big exception here is Columbia, which releases their new essay questions and opens their application in May (in past years, only ED - Early Decision - and J-Term candidates could submit apps right away, and the regular-round September Start candidates had to wait a month or so before that app was opened). The Columbia rolling admission is totally different from any other school and it's easy to get confused by how it works in comparison to the standard process that schools like HBS and Wharton follow. But, Columbia traditionally kicks off the admissions season in this way.
Many other schools release essay questions in June but they often don't open their applications for submission till later. NYU is the worst in this regard: they typically do not release questions early, and they usually don't open the application until August.
Essay questions do change year to year, sometimes radically. NYU is an exception here; they haven't changed their questions significantly in ages (not guaranteed they won't this year but it's unlikely).
If you're eager to get started, the one essay you could safely tackle today would be the standard "career goals" one. You could use Columbia's essay 1 question from this past season as a good full-bodied version to start with; other schools have different variations on how they ask it, but if you do a good job with the Columbia one, then you can adapt it fairly easily to others.
We don't recommend doing too much pre-work on any other essays, since it's likely you'll need to start over again once the current season's questions come out.
There's no real advantage to submitting early at most schools anyway (except to avoid server congestion on deadline day). Harvard and Wharton have specifically said this is true in the past. Columbia again is the exception here; because of their unique rolling admissions process, there is a BIG advantage to submitting early there, so if Columbia is on your list, you should try to submit during the summer. Stanford and UCLA have mentioned that they like getting applications early but we don't know how big an advantage it actually is. If the GSB likes you, they're gonna admit you regardless of when the app comes in.
(We're also not sure what was meant by when you can "drop my application off" -- all top-tier schools in the US require online applications.)
EssaySnark will be reviewing each schools' essay questions on our blog as soon as the questions are released. If we can help in other ways, let us know!
(ETA: The HBS 2+2 (for college seniors) program application typically is released in April with a deadline in June. HBS is changing the 2+2 app process this year, however those dates will still apply.)
