Can you put module layout files within the Module's code directory in Magento 1.13? This is as opposed to putting them in /app/design/...
In short, no, you can't. But it is the way Magento 2 is heading...everything in the single module directory.
If you are creating a distributable module which has it's own layout file, put it in
/app/design/<area>/base/default/layout/ and it will take effect regardless of the theme that is used. Just please don't do this if it's custom development work for a one-off project.
No, not natively, but it is certainly possible - and really quite easy at that.
There's another answer here that outlines how to add another directory to be read when parsing layout XML files (and in order of preference).
You could make couple of adjustments to make it also read from your own module directory.
No you cannot. You will be able to do it in Magento 2.0.