Obviously, Google is providing a free service, and one of the ways they afford to do that is to collect and sell information about its subscribers, which it does quite expertly. Most of the information they currently gather is through GMail, which provides a steady stream of email that is scanned for keywords. Google then distributes targeted advertising according to what they think your interests are.
Private companies use Google apps to share data and provide computing services. In order to limit access to the data, and restrict interaction with outsiders, commercial ventures often will choose a private cloud model which is more secure.
Unless there is highly sensitive content saved in the Google Docs database, I think the risk is pretty small of hacking from outsiders. At this time, the use of targeted advertising is very limited in both the calendar and docs. It is easy to set up a folder for each student that is restricted to the teacher and student. Individual documents or folders can be restricted reliably by email (account) address. Once set up, the student does not even need to access email, but each student must have a Google account.
One of the great things about Google Docs is that it is accessible from a variety of platforms, in contrast to Microsoft's offerings. Apps are available for both the iPad and Android devices that allow those devices to modify documents stored in the Google Docs database.
I am currently using Google Docs to share content among teachers. I find it exciting that you are thinking about using an electronic dropbox. I think it will be a while before schools in this area would be comfortable with such a plan, because there is a large percentage of students without computer access. As more of them have phones with data plans though, it is more realistic that some of those students will have access away from school. It also has great features for document collaboration.
Unless the information stored in the database is extremely sensitive, I would not worry about the security of Google Docs. Just understand that it is a free service, and there will be trade-offs. Greater security is expensive, and until school find themselves awash in cash, Google has a wealth of tools with adequate security.