I've actually been on a team that did this. "For the next two-week iteration, we're going to change the process in this way, and we'll see how that works."
Process doesn't have to be a static thing handed down from on high. Hack it. Make it work for you. (One of the agile approaches to software is "pay attention to pain". That is, when it's hard to change the software in some way, that's telling you that it should be refactored so that it quits being hard to change it in that way. Well, the same thing works for process. If the process is painful, pay attention. It's telling you that you need to change the process.)
Yes, that's exactly what Lean in software means (and you'll note that it has a very close relation to Agile principles -- Lean is pretty much Agile with more focus on measurable output and several decades more of development behind it -- both for the good and the bad [there's a lot of depth in the literature of practices and how and when they work to draw from for your circumstances, but there's also breadth and depth in the buzzword snake-oil].)
Practices like continuous integration and unit testing traditionally fell into that category. I would say these days that writing unit tests is so common that it doesn't feel like it came from anywhere. Just that it always existed.
The ossification of Agile into a set of cargo cult practices divorced from the particular teams and their needs flips Agile into exactly what it was a reaction against.
In my experience the main thing to keep in mind to improving any system is to:
1. See what state you are in 2. Figure out what new state you would like to be in 3. Guess at something that will help you move into that state and then follow that guess. 4. See if that move helped you get closer to that state 5. goto 1
People label those steps with lots of names depending on what industry they work in but in the end that is the easiest way to get better.
(2) The only relation anything "Agile" has to do with manufacturing is that the Agile approach to software is close parallels to Lean methodologies in manufacturing (which have circled around to create Lean software methodologies.) But Lean, while it is an approach to improving productivity, is pretty much the opposite of doing so at the expense of the creativity of the line workers, as they rest on the combination of defining and measuring value and harnessing the deep knowledge of the workforce to propose, implement, and test methods of improving value produced.