If I double tap the home button I will get the recently used apps.
I have an app which I must keep on but I don't want other apps to send any data in the background and increase my data usage and my billing amount.
If an app is currently running will it 100% be in that group?
If an app is running, it will always be in the task switcher, but just because an app is in the switcher does not mean it is currently running. There really is no way to say if an app is running with 100% certainty other than to open it (with the exception of audio apps, which if they are playing audio, they are running). See Apple's page on multitasking.
The general rule about background applications is that they get five seconds of background time. An application can, however, tell that it's performing an background process and that way extend the background time to about a maximum of ten minutes. This makes it possible for applications to start a download when it's active and continue it if put to background.
To this there are a couple of exceptions where applications wont get suspended when put into the background.
There are also all applications that you allowed to receive push notifications that use data. This together with the built in apps people already named. You can of cause choose to inactivate automatic mail fetch, turn off the iMessage functionality and disconnect from iCloud.
Macworld have a nice article that explains this and the different states of applications.
You can also dive deep into the App States and Multitasking at iOS Developer Library.
It depends from application to application. One such example is TuneIn Radio for iPhone it uses internet to stream radio even when the application is in background state. If you want only your application to use internet you can close other application which are in background state by holding on one of the application icons to trigger wiggle mode. Once they start wiggling you can remove unnecessary applications :)
P.S : iPhone default applications such as Email client, iCloud service, iMessage also consume internet bandwidth. So make sure you disable all if you only want certain applications to consume bandwidth
There are apps like WhatsApp which can receive updates (and therefore produce traffic on your data plan) even if you close them in the overview you mentioned.
I don't think there is a way to block this behavior without disabling data usage completely. Your could change the settings on the most of these apps to just update manually, but there is no switch to do so with all or a range of apps at once.
