How do I investigate potential unauthorized access to my iPhone?

we are not apple

these forums are user helping other users like yourself, it's sort of going to a physical apple store and asking other customers their advice

someone accessing your phone is not really a thing, but them accessing your account could be

it would usually because you provided them with access though

read

Identify legitimate emails from the App Store or iTunes Store - Apple Support

&

Recognize and avoid social engineering schemes including phishing messages, phony support calls, and other scams - Apple Support

&

Phishing - Wikipedia

if you feel they might have taken access to your account then read

If you think your Apple Account has been compromised - Apple Support

I am unclear why those images are related to your question.

From what was posted, there is what looks like a cellular data problem that led to trying to use SMS or RCS, and that attempt also failed. That can happen when there is a network connection issue or related communications failure, or problems with the cellular carrier or related details.

Running in low-power mode also shuts down various functions. See if things work better when running in full power, and also see if things work better when in a different location with different cellular coverage; if this is location- or tower-specific.