Hi magoodfel
Welcome to the Apple (user-to-user) Community.
Wondering which iOS version is installed on your iPhone (eg: iOS 16.2 is the latest)
and
which Mac air Model (year)
and MacOS version (ie: if compatible: latest is Ventura MacOS 13.2.2)
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Re: attempt to sync photos from my new iPhone 12 to my Mac air
Wondering which method you were trying to use to sync
(eg: ? can you use the Finder on your MacOS version)
How to: Sync files from your Mac to iPhone, iPad or iPod touch – Apple Support
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Re: "the screen to trust or not trust pops comes and goes super fast".
Wondering which app the screen is in:
If your iPhone is not working right, you may want to try re-starting it:
Restart your iPhone - Apple Support
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Re: "If I reset my network settings, what does it mean to reset to factory defaults?"
You might want to scroll to:
"Reset Network Settings" in the article below for all the info:
Return iPhone settings to their defaults – Apple Support
also:
If your iPhone or iPad won't connect to a Wi-Fi network - Apple Support
says:
"This also resets Wi-Fi networks and passwords, cellular settings, and VPN and APN settings that you've used before".
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if needed:
iPhone - Official Apple Support
(NB: There's a "Get Support" link at the bottom of their webpage)
All the best :-)