[Lizard]

On the Mac, you can set the project for an action in the inbox without dragging it to a project.
There's a preference in OmniFocus for Mac labelled "Clean Up Inbox items which have:". This is the preference that controls when OmniFocus will move actions out of the inbox on your behalf. This setting is synced to all your devices.
I think you probably have it set to 'Both a Project and a Context'. It sounds like you're not really using contexts, so you probably want to change this setting to 'A Project' and then sync all your devices.

Does that help?