you can install google apps, or CM12.1 rom, not quite stock android.
Don't think you can replace settings, without installing a custom rom (CM12.1)
should be able to install android calendar and email, but amazon may have taken steps to block those from working.
Sorry for the ignorance but I've been out of the Android world for awhile: can you download the Google apps from the Play Store? Or do I need to get the apks and sideload them?
How is the CM ROM? Last I heard it was still somewhat buggy on the new Fire.
Thanks!