Never really thought about it but now that you bring it up I guess I always do the right foot first. Not sure why, maybe because I'm right handed so starting on the right side 'feels' like the correct one to start with?
No specific order really. I sometimes wear an elastic sleeve on my permanently swollen left ankle after my bad sprain 2 years ago, I might put the sock on over that first, but not an always thing.
Generally, left foot first. But that's only since I had hip surgery a couple of years ago on my right hip. As I'm getting "better" (flexiblity-wise) it can vary, depending on how flexible I am that day.