You can set up a default OBS Department on the Blueprint and template, so once the users select it, it would automatically carry the Department and Department OBS over from the template.
For Ideas this is currently not possible.
We suggest you bring this up on the Product Management call to prioritize.
One thing that could help in the meantime that the functionality is not yet there, is to create a process on create for Project and Idea that would either 1. fill in the Department based on user's department (more complicated) or 2. Send out a notification to the user reminding them Department is unfilled