Is mobile internet possible with disabled data transfer to android?

Question

Android

Hello.
In general, there is a smartphone on Android 3.4.0, it has a SIM card with a beeline service "Internet for a day" (unlimited for a day for 19 rubles, activated at the start of data transfer and valid until 24.00), there is wifi and mobile Internet is sometimes needed once a two or three months, because the transfer of mobile data in the phone is always turned off, it turns on only when necessary and turns off as soon as it becomes unnecessary (because, among other things, it also eats charging).
Today I ordered a detail (noticing that the money flew somewhere, which was not there before), in the detail I see "Mobile Internet" at 5-50 kb - almost daily (except for a couple of days), the money was withdrawn in 4 days, but what prevents the operator billing and the rest?
Main questions:
The standard "traffic control" on android says that there was no data transfer. Can he lie?
The operator's specialists say that applications can allegedly be updated when data transfer is turned off - is this possible? Updating applications via mobile Internet with data turned off and wifi turned on?
If this is not possible, how to prove it to the operator? The "Mobile Internet" is fixed in its detailing, how to argue with the detailing?
PS: the garbage started from the new year, there is no Internet in the December detail, although the smartphone is the same, the applications are the same, and the operator.
UPD: Android 4.1.2, wrong.
At the moment, the money was returned, but the mobile Internet is already in detail for today and it’s not clear, they returned it because they realized that it was their mistake and they wouldn’t do it again, or they returned it so as not to get involved because of trifles and in the future you can wait for a repeat.

Answer 1

Lies, zvizdezh and provocation.
This is the jamb of the operator. Or perhaps you clicked on data transfer by mistake and immediately turned it off. This is where the package could slip through.