My clumsiness kicks in a few days before I start menstruating, and carries on til about day 3 or 4. I know for sure I'm coming on when I start opening cupboard doors into my own face, knocking things over, backing into stuff in the store, etc.. While I'm no Margot Fonteyn the rest of the month I'm far less clumsy in general, which is why I think it is due to hormone levels changing. It seems to be a spatial awareness thing, my hand-eye coordination doesn't seem to suffer.
As for the immune system, I have been taking Vitamin D and Calcium supplements for over a year now. I started taking them because I read that they might help with mood swings. They did - they had an even more pronounced effect on breast soreness, reducing it dramatically. Since taking those supplements I find that a sniffle or sore throat around the time my period is due is no more likely to develop in to a full blown cold than it would at any other time in my cycle.