Apple Silicon machines cannot boot from external storage. While it may look like they do when you choose an external macOS volume, behind the scenes parts of its boot components are being copied to the internal drive to make this work. It’s unclear whether this mechanism will ever be usable by third party OSes, for technical reasons.
Instead, we recommend using the UEFI environment only installer option to install only a UEFI bootstrap to your internal drive. This only requires around 3GB of disk space, and it will then automatically boot from any connected USB drive with a UEFI bootloader. Note: installing the Asahi Linux desktop images to a USB drive automatically isn’t supported right now, though if you’re adventurous enough it’s not terribly hard to do manually :-)
Sadly no, and it's unlikely for any OS to be livebooted on Apple ARM systems (effortlessly, like on x86 BIOS/UEFI). macOS does this cheat of "copying bootfiles it needs in internal storage", and it'll be likely that Asahi (and any other OS) would need to set up a permanent partition just so they can pretend to boot up external drives.
M1 boot sequence requires some code to be installed on the system. Once the execution jumps to the bootloader the rest can be run from anywhere, such as USB stick.
It still requires adding a new partition with bootloader and some other files required by Apple for successfully verifying the signatures on everything, so it's not a "live mode".
I’d love to try it for fun on a computer that is not mine…