How to prevent 1P's icon from appearing in the Dock at startup

The easiest solution would be for your design team to add two checkboxes: hide from dock and hide from menu bar. Apple does this with control panel items, for instance, in the menu bar, and it works well. You can always force-launch 1P and go into settings to find it. Default to showing both, and when the user un-clicks (at least the first time), show a dialog box explaining how to get back to default behavior.

This is not rocket science. Lots of apps and Apple accessories (bluetooth, clock, sound, etc.) have this kind of behavior.

If I'm not mistaken, you used to have an option to keep alive the 1password mini process after quit; that was perfect.