Is there anything from stopping me from taking a codebase licensed under the GPL/AGPL, modifying it, and then selling it as a hosted application for a monthly fee as long as I make my modifications available to anyone who asks?
At what point does a derivative work become a new work?
You can sell any GPL software to anyone for any price you care to ask. You still have to provide GPLed sources though.
If you are asking how much (little) you can change in GPLed code and legally start hiding the sources from people you redistribute it to, the answer is that you have to change all of it.
You're completely within your rights to do that (and in fact it's exactly what the GPL was created for - to ensure that the users of a piece of software have access to the source code).
(Note that you'll need to make the entire source code of your derivative work available, not just your modifications.)
A derived work never becomes a new work, as long as it still contains pieces of the original work. But that doesn't matter in this case, since you're obeying the (A)GPL (in both letter and spirit).
Yes you can do that, but remember that you will FOREVER have to provide ALL your sources that are based on the original work, or that inter-operate closely with it (figure out what linking means for your language of choice). You don't get off by providing just the original sources you modified.
For-sale GNU licensed software includes things like Tivo as well as GNAT.