So I'm logged in as "Amber," and I have privileges to log in as any user in the system. If I were to log in as "Jessica" and open the developer console, would I still be able to access information about the profile "Amber" with Apex code? Is there any class for this, or is all the information about the previous user you logged in from completely lost?
The Developer Console's API, like all other tools, is tied to a user's Session ID. So, when you log in as a separate user, the Developer Console would get that user's Session ID, and be restricted to that user's permission. This even includes logging in by way of the "Login Access" mechanism for administrators. This means the user you're logging in as would need to have at least View All Data to even load the console, and it would load their configuration (e.g. "Jessica") instead of the user you're logged in as (e.g. "Amber"). None of Amber's data would be visible to Jessica.