TNTT Is it possible to automatically switch to Owner profile when I lock the phone (press power button) while using a secondary profile?
No. This feature is mostly for different "identities" using the phone, i.e. work TNTT vs. personal TNTT. So when you're at work, you use your work profile. But if you want to keep the profile active, just pull down from the top of of your phone (like when you want to see notifications) twice, then tap the user icon to switch back to owner, or some other profile.
Keep in mind you can have owner + 3 other profiles active at the same time.
TNTT Is it possible to "sync" device settings between profiles?
No. Profiles are designed to be very isolated from one another. They cannot talk to each other. Not even the owner profile can modify settings in another profile because of this isolation. Some settings are global, however, such as network settings.
TNTT Do Google Play Services and Google Services Framework consume more battery life, the more I enable them on each profile?
Yes. Each instance of Google Play Services runs independently of each other. VPNs work the same way. If you have 2-3 profiles, each with their own VPN and GMS instance, you will definitely see a lot of battery drain.
Keep in mind each app is sandboxed. Profiles only add a little extra isolation from other apps. I'd read more about profiles on the GrapheneOS website here: https://grapheneos.org/features#improved-user-profiles.
TNTT I guess I can't put a widget of an app from a 2nd profile in the Owner profile but can I have a shortcut icon of an app?
No. Also because of the way profiles are isolated from one another.
TNTT I tried changing the search engine to Startpage -that isn't suggested by default- but couldn't. I don't mind using ddg but can I change it?
you can change the default search engine by following these instructions