as long as the ID of the attribute is the same between the project object and the idea object it will copy as expected.
The value should remain the same.
If the ID is different, you need to create an object mapping XOG to get the mapping right on conversion.
https://www.ca.com/us/services-support/ca-support/ca-support-online/knowledge-base-articles.tec1998952.html
As for the manager field, when manually converted it is the user who converts who is set as the manager. (because lookup shows logged in user)
What behaviour do you want to happen? (idea manager becomes project manager??)