I think it would hold up fine on a run but I find the heart rate sensor isn't as spot on with how tightly (ie somewhat loose) I wear my link bracelet. I could remove a link to make it tighter, but at that point if I've already separated the links to do this, its only a few more seconds to swap to another band. At this point I have a rhythm to change bands and can probably go from one to the other in 10 seconds... 15 including the time to take it off and put it back on my wrist. It becomes super easy very quickly.
I worry that lifting weights (or doing pushups or anything that causes flexing in the forearm) could put a lot of stress on the connections between links. Since I've got a few sports bands, and paid a fair amount for my link bracelet, I take the cautious route and just use the sports band for working out, running & biking.
EDIT: And to answer your other question, I don't think sweat is a problem for the link bracelet... I've been wearing it during some really hot humid weather and its gotten covered in sweat several times. A damp cloth and it all wipes away.