there are two types of the Pico chip, RP2040 and RP2350. they’re used in 1 and 2 boards respectively. W boards of either kind use the same chip, but have a wireless chip added to the I/O.
what this means is that any firmware built for 2040 will run on a 1 or 1W, any for 2350 will run on 2 or 2W. of course, wireless features will not work on non-wireless boards, and firmware is (usually) built wireless-specific because there is no (good) way to determine at runtime whether or not a board has the wireless hardware added.
so, for a 2W, any firmware marked for use with 2, 2W or 2350 will work. firmwares marked 1, 1W or 2040 will not, simple as that
broadly speaking, the instructions for what you’re looking for (installing a new firmware on your fresh 2W, i assume) are to just hold down the BOOTSEL button on your board, and while you’re holding it, plugging the USB cable to your computer. the Pico will show up as a removable mass storage media, and you simply drag the firmware’s UF2 file to it. whatever happens after that depends on what the firmware is
i recommend using UF2 Loader release so that you can swap different firmwares stored on your SD card directly from the picocalc without needing a computer
as for PicoMite specifically, i have no idea what the latest version is anymore, since the codebase seems so splintered. i’m not personally following it myself