Can police officers know what you are doing on your mobile phone?

Yes they have a machine that they can see what you were last using your phone to do. If they pull you over for texting and driving and you say you didn't they can tell anyway

Call records are not easy to get but the police can get them from the phone provider with a warrant. 

I suppose they could "tap" the phone. The technology wouldn't be that difficult. That would require a warrant also. 

And of course if you hand them the phone they can look on. 

 But they can't do anything without either your permission or a warrant. 

If they have the equipment with them to tap into it they can.

Not without a search warrant.