Android - How can I know which apps are currently connecting to the Internet?

OS Monitor does that. Shows the name of the app, and the IP address of the connection (including a WHOIS with a map when you click on it).

EDIT: I should mention that the sync icon TYPICALLY indicates a system-level sync. Do you have every item (GMail, Contacts, Calendar, Facebook) explicitly turned off in the settings?

PhoneUsage keeps track of how much data you use each day, and gives a nice display of which apps used how much data.

Here is an example of what mine looked like this month:

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NOTE: It only starts tracking after installation.

Check out SPB Wireless Monitor. It can give you reports by date range of which apps have used data, how much data they've used, and breaks it down so you can see wifi data vs. data-plan data.

There's a new app, My Data Manager, which costs a fraction.

You can check that out as well. It does basically the same thing, though will also show you roaming data. Also, this one allows you to enter a monthly start date so you can align its data tracking with your billing cycle.