iPhone regularly shows "iPhone not activated..." message
Have you contacted your carrier, as suggested?
First Question
I believe that this page may help your case. From what I know, iMessage and FaceTime will not work on non-activated devices. You will need to first determine whether your device is really not activated or whether the message is randomly appearing although the iPhone is activated. Follow this guide to check.
Second Question
To address the second question, I am assuming that data refers to your iPhone's internal storage and not the cell carrier/mobile plan related data that allows you to browse the web.

Removing the SIM card does not delete the iPhone's internal storage, thus you do not need to worry about data-loss during the procedure because storage is not related to activation. The only case in which this would occur is if the device is fully reset.
You should have a current backup of your iPhone. You mentioned that iMessage and FaceTime are operable, so I assume you back up to iCloud or at least to your computer via iTunes. No matter what you try, a full backup will always come in handy if everything fails.