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It sounds like you have a multitude of problems.
Have you tried a FDR [factory data reset]?
Find a local repair center and/or contact Moto.
Your battery saver in "Smart Actions" is doing the data restriction and the message when the battery falls below a certain percent of battery...
For the most part, profile pictures downloaded from FB or elsewhere will be low res.
If you are using the stock launcher go to Apps/Widgets, Favorite Contacts. Drag it to any empty space on your screen.
Enjoy your new toy!
Probably one or more apps running in the background.
Make sure you charge it completely.
Just found this online yesterday. Hope it helps.
Keep making random patterns about 5 times or so and it should lock you out for...
I have been using the free version of AVAST and it works fine. I am also using it on my laptop with good results.
What ROM are you on?
A FDR is always recommended after a major update.
Are you using the "Gmail" app or the "Email" app?
It sounds like the last time you viewed the calendar you were in week view. Unless you change the view, every time you open calendar it will show...
I use a screen pattern lock with the bluetooth bypass and have no trouble with what you are describing.
The person you purchased it from online may have rooted it and removed/frozen parts of the OS.
There is a time limit [I don't know what it is] on the free version.
Have you done a FDR since the update. Doing a FDR can fix many problems.
If you're waiting for Motorola for a fix, you may be waiting for a long time. Routers have come down in price. I think it's time to buy a new one.