Suspicious Calendar Events, Spam Phone Calls, and Outgoing Messages—Has my iPhone been hacked? Has my iPhone been hacked?

Part I: The FaceTime Call

Yesterday evening, after turning on Do Not Disturb mode for the night, my phone begin to ring (an incoming FaceTime call) with a ringtone I have never used. The caller was not on my favorites list (so the call should not have made noise since I was on DND). That caller is my co-worker at the office, and claims to have not called me (and indeed his phone has no record of the outgoing call).

Part II: The Outgoing Text Messages

Today, I noticed an unsent text to several phone numbers I do not know (and a few I do know, some of my friends) with a political message I do not support. This wasn't me sending it, so what happened?

Part III: The Suspicious Calendar Events

As detailed in this Ask Different thread, I'm having spam calendar events appear as invitations.

Conclusion

I have no idea what is going on with my iPhone! Have I been hacked? What can I do to resolve this problem with my iPhone?
Most likely, your Apple ID was hacked. 

Any device signed in to your Apple ID can send texts, change notification sounds for specific contacts, change contacts phone numbers and more.

Go to appleid.apple.com to change your password, and change passwords on any sites that you use the same password on, as those are at risk as well.

If your password has been changed by the perpetrator, you will most likely have to contact Apple support, as well. 

To prevent this from happening in the future, I recommend adding two-step verification to your account, so every log in has to be approved via a code on one of your devices or phone numbers.