1: Can I use a Linux boot stick to check the desktop windows partition for viruses and/or rootkits? Yes, you can. Examples include Bitdefender, AntiVirus Live, and numerous others.
2: How is a casper-rw file different from a casper-rw partition? Same thing, a casper-rw file is treated as if it is a writable partition, and is used to store your changes and additions to the filesystem.
3: What are limitations of a Linux boot stick with persistence that I might (and most probably are) not be aware of? They boot slower, and most LiveUSB creation tools (one example), if they offer persistence, are limited to a maximum of 4095 KB in persistent size. See https://www.pendrivelinux.com/what-is-persistent-linux for an overview and, for an explanation of how to exceed that normal limit, see https://www.pendrivelinux.com/create-a-larger-than-4gb-casper-partition .