The red stringy slime just sounds like poop. I would think the events are unrelated.
As to why one would split and another not ... The way I look at it is, they'll just do it whenever they do it. Your other anemone could split at any time, or, who knows, it could be a year or more before it splits.
If they are clones to begin with, then either of them have the same disposition to splitting. Joyce Wilkerson says in her book that the only sure way to predict whether a specimen has the potential for splitting is to know whether it has ever split in the past. This suggests that some never split, but in your case it sounds like your big guy can in fact do another split (whenever it gets around to it).
BTW, there are two different forms of E.quad, a shallow water version that is colonial in nature and a deeper water version that tends to be larger, and solitary in nature. I think there has been speculation by a few that the deeper water versions do not split (or rarely split), because they are usually found singly on the reef whereas the shallower water versions are always in a colony of many anemones (clones of one another). I don't know whether these two forms are considered different subspecies or whether it is simply the different environments that produce the different behaviours.
There is a fellow on RC who has something like 60 rose BTA's in his tank. (Personally, I think that's nuttier than a fruitcake, but, ... to each their own!
). I think his name is JBF16Falcon or something. You should do a search on RC if you're interested, because I seem to think that he believes he has discovered the secret to encouraging splits, so maybe he can shed some light on the topic...