So I missed some of your questions.
It depends on the type of modem. Many times you must reboot the modem, wait for it to come up and then plug your pc in. A modem tend to lock to the mac address of the first device it sees so if you just swap them it will still only want to talk to your router mac address. The only reason for this test is to eliminate the router as the cause.
A linux USB stick is very similar to a windows install image where you can use it to fix or install the OS. I wish I had a exact link for you but it is generally a debugging tool that has all kind of other abilities if you know what you are doing. It will boot the linux os from a USB stick without writing anything to your disk. It generally has a browser installed so you can just run speedtest like you do in windows. I have a really old one but I use it so seldom I have been to lazy to get a newer one. Likely I need to fix this because the idiot secure boot stuff and lack of modern drivers is going to bite me. Maybe someone else that sees this thread has a recent link.
The only real reason you use the linux image is to see if maybe there is something wrong with the windows install. It unfortantly does not tell you what is broken and many times people just reinstall windows and hope that it was some software that was installed or some setting that was changed. Generally if you had another machine that test at full speed then you come to a similar conclusion except running the linux image confirms that the hardware is good.