avast! has to be installed in the target operating system to work correctly (with all available features). While its folder theoretically may be on a USB drive (though it probably won't be exactly fast), it needs some records in Windows registry, some files in system folder, etc.
Regarding fixing other computers and the corresponding registration - there's the "serviceman license" for avast! BART CD, which is meant exactly for this purpose. But of course, it comes at a significantly higher price.