Why / When do I have to log in to Google? Installed MyDocs, I can select my Google-account and see listings of my documents, but to edit something I need to enter a password.

Why is this?  When does it happen?
(I am constantly logged in to GMail)

I use Android because all this Google stuff is supposed to be integrated!



Edit: How can i know if an app is really integrated, and won't require separate authentication.
You're right that Android is supposed to be integrated. Well it is, but it doesn't happen magically, developers need to make it so.
This specific app is a beta and it was last updated more than a year ago. It's a safe bet that the developer(s) abandoned the project. 

I recommend that you install the official Google Drive app. It does not ask for your password and it is regularly updated. 



Applications that can use the integrated authentication system require various permissions in the "Your Accounts" category.