Okay, so, time for me to give a bit of an update. I was able to resolve the issue. What I had to do was go back onto my old computer, create a new folder in the missions game directory, open up all of the files I can't find on my new computer and re-save them in the new directory, and hey presto, suddenly there were .mis files. I then transferred them into the missions file directory on my new pc and they work fine!
However, at the same time as I got these files to work in the game on my new computer, a new problem appeared. Even though the game was running absolutely perfectly just two days prior to solving the mission files issue, the game has now developed a bug that I have no idea how to fix:
Basically, when I open the game, the Windows cursor is not replaced by the in-game cursor. The Windows cursor still allows me to navigate the game's menus no problem, but when I load any mission into the FMB I cannot click on any objects that are on the map, it simply won't let me select anything. Additionally, when I launch a mission from the QMB, the cursor doesn't disappear at the start of the mission. This limits my ability to pan/tilt the camera, as the cursor moves around the screen when I use the mouse to look around, and when the cursor reaches the edge of the screen, it won't let me move the camera any further. Lastly, the esc key on my keyboard also does not work during missions, but works fine on the game's menu screens. It's truly bizarre. I have checked the [rts] section of the conf.ini file and mouseUse is set to 2 as it should be, so that's not the cause of the problem, either.
If anyone has any ideas on how to fix this new problem, I'd be so, so grateful.