Does 'find your iPhone' use a lot of internet? My mobile data is running out ridiculously fast and I need it to slow down!

If you have iOS7 you can go to Settings > Cellular and see some information on the amount of cellular data used by Apps and by System Services. Within System Services are details including Find My iPhone. As noted above, Find My iPhone data is minimal.

You should shut down all the apps running in the background and then reset your phone. Double tap the home button to open the multitasking drawer (iOS 6.X) or the multitasking screen (iOS 7.X) if you are on iOS 6.X press and hold one of the app icons and wait for the icon's to start to jiggle, just like when you want to delete an app. Tap the little x that appears in the upper corner of the icons. Repeat until all the icons are gone.

If you are on iOS 7.X once the multitasking screen is open swipe up on all the app images until they are all closed. Tap on the home screen to close.

Then press and hold both the Home and Sleep buttons at the same time and wait for the Apple logo to appear then release both buttons. The phone will restart itself.

If this doesn't help try doing a Reset All Settings reset, you will not lose any data or content or media. But you will have to reset any setting you turned on or off in Settings. This will reset all settings back to factory default. Settings => General => Reset => Reset All Settings.

No, Find my iPhone uses very little data. You have one or more apps that are using data.