I have a 16gb flash drive which I want to contain a portable Linux install. If I could just install Knoppix onto the flash drive, and have that work on any computer, I could do that. Would that work, or are hardware specific things set when you install?
While "work on any computer" is a bit too broad, it is possible to build and configure a system so that it works on as many hardware configurations as possible — after all, that's what projects such as Knoppix try to do, have a kernel with lots of modules available and run clever startup scripts that try to tweak the configuration during the boot process.
Even if, in some cases, it may fail, it's possible to tweak some boot options to workaround known issues (again, that's what Knoppix does with its "cheatcodes", inserted at the isolinux prompt.
This said, even if it covers most cases, I'd still keep some way to exchange data, just in case you're unable to boot it in some computer.
(More technically, hardware-level settings are mostly when you configure the kernel. But many distros just ship the same binary kernel for all its users who prefer not to compile their own kernels. Some hardware-dependent logic may be also set in the configuration files, but that's not as drastic as lacking the correct kernel driver for some device — it will probably just boot but show some error.)