FS-UAE includes a portable mode where it can be completely self-contained within a single directory structure.
You can generally upgrade to a new version by simple extracting the new version on top of the old one.
FIXME: Write about old files being left around, how to clean up, and/or how to upgrade "cleanly".
You can combine the portable versions for all supported operating systems and CPU architectures. Just extract every portable version (the ones you want) on top of each other.
It is important that when you upgrade to a new version, that you upgrade every portable version you are using at the same time, since some files are shared.
The portable version will still store temporary files in the system-provided temp directory. This is the only exception where FS-UAE (Launcher) writes files outside the portable directory (except also if you specifically use floppy / HD media from outside the portable directory of course).