I make pickles similarly, but not quite the same way: I usually make half-sours, as even the 5.6% salinity of that recipe would be way too salty for my family's tastes. I boil the brine before using it, letting it cool before adding it to the cucumbers. I use a clear glass jar rather than something opaque like stoneware, which might change the mechanics a bit. The biggest difference, though, is that I add a slice of bread, which really gives the fermentation a head start. Usually, the pickles are ready in two days, three at most. Perhaps it's this shortened time period, but I've never encountered any scum that needs to be skimmed off. Sometimes, if it was a particularly hot day or something, there will be mold, either on the bread or on the liquid; if this happens, the pickles are thrown away — it's just not worth the risk.
That said, scum does not equal mold. If it's anything other than white, and/or if it's fuzzy, it's mold, and I personally would not eat whatever it occurred on, unless of course it was rotten moldy milk blue cheese. If it's really just scum, then it's probably a cosmetic issue: if you don't skim it off, the brine might turn cloudy and unappetizing.
I would say you could use it once, and probably just to make refrigerator-type, quick pickles. The problem is that you have no way to determine acidity or salt levels once you remove the previously pickled items. Of course it should look clean and clear. If you were previously doing a lactic acid fermented pickle, some of the liquid could be used to jump start your next ferment.
Best Answer
If your solution grew mold, it would indicate that either the container wasn't sufficiently sterilized before sealing or (probably and) the solution wasn't sufficiently acidic. This can usually be ensured by following a recipe that is known to be good, which should have factors like acidity and water activity accounted for.
As for kahm yeast, it seems that this occurs more frequently on cultured vegetables which would spend some of their time in a less acidic environment (while the culture develops and acidifies). I suppose you could culture it and test it, but when in doubt I'd just toss it.