I'm not really sure what you mean. If you have a live usb stick and boot a system of it, like fedora or mint, there is usally an installer to install the system onto the pc. this does not alter your usb stick though. you can install as often as you like.

I don't get the download part either. the live usb stick is the installer. no need to download additional files.

Remember the live usb environment is not persistant. after every boot from it, all files you have created inside the live environment will be gone.

If you’re asking whether a live-disk usb is reusable the answer is yes. You can re-use a live usb as many times as you want (as long as you don’t partition it by accident during your install)

If you’re asking if you can use a usb as your file system, the answer is also yes, with the caveat that definitely do not want to plug it into a different computer.

I THINK what you're asking whether the changes to a Live USB system ALSO get installed down to your machine if you make changes while live.

I do not believe that any do this anymore for a number of reasons, mostly the complexity of tracking those changes through install steps.

I assume you want:

• a persistent Linux USB stick (basically you can take your OS and your files with you on the go, and everything is stored on the USB), and
• a Linux install on a desktop/laptop, which you created using your persistent USB from above.

I believe this is possible by just creating a Live USB stick with a persistent partition, you can store files on the persistent partition, and still have all of the installation tools you usually find in a Live USB. On Windows, this can be done with Rufus. Or you could use Ventoy as described in this guide.

Everybody seems to have a different idea of what you mean by your question. I interpreted it to mean that you've got files on your USB stick that you don't want to lose and you also want to be able to boot off of it.

If I'm right about what you're asking, the short answer is "yes". It'll require some research to get all the pieces put together right. The specifics will probably depend what bootloader you put on the USB stick, what filesystem you have set up on the USB stick, whether the stick uses an MBR or GPT partition table, you're booting via legacy BIOS boot or EFI, etc. But here's a place to start: https://wiki.archlinux.org/title/Multiboot_USB_drive#Using_Syslinux_and_memdisk