Basically you shouldn't download anything as Run, but save to your hard disk, or you would be on-line whilst installing an application and if you have a problem, you haven't got a copy of what you downloaded.
I think you just said what to do.
Your desktop is a location on your Hard Disk, however I prefer to have a specific location for downloads, like C:\Downloads (a folder called Downloads on your C:\ drive) this makes it easy to find later as your desktop may be different in safe mode.
So if you saved the uninstall tool to the desktop boot into safe mode and double click on the icon for it on your desktop and that will run the avast uninstall utility, reboot when complete.
No your problem will be since you didn't save the avast home version to your HDD you won't have a copy of it and you will need that to reinstall. This could mean being on-line without protection whilst you download and save it, which isn't very good.