Using carbon with a skimmer for ozone usage

[Muffin87]

I just bought this Sander skimmer, blueprint below, to use as an ozone reactor, because it was 80% off, 26 inches tall, very small footprint, and ozone-safe (Sander also produces ozonizers in Europe).
I've read Randy Holmes-Farley's articles on ozone, but it's my first time using it, and I'd like a couple of pointers on GAC usage with the skimmer.

1. GAC to clean air leaving the skimmer i.e. putting carbon in the skimmer cup
   Issue: Is the ozone gonna shut down skimmate production to the point that I won't need to worry about water/foam entering the carbon-filled skimmer cup? Otherwise, how do make sure the skimmer doesn't produce skimmate?
   
   Solutions?: Should I stop the skimmer from producing skimmate by increasing the rate of the ozone relative to the air pumped in? By tweaking the flow rate?
   The skimmer can be used with two wooden stones as well (see picture), I was thinking I could use less porous stones to produce fewer, larger bubbles, which should suppress skimming theoretically. But isn't that also gonna decrease the amount of ozone that enters the water?
   Alternatively, I could fill the skimmer with bioballs?
   
   
2. GAC to clean water leaving the skimmer
   Is it a good idea to pass the water leaving the skimmer through a tiny 350 ml carbon reactor made completely in acrylic? i.e. Sera multifil 350 pictured below.
   I was watching BRS video about ozone, and it was suggested to just put a media bag/filter sock filled with carbon on the skimmer output. I'd like it to be a bit neater. My cabinet is cramped and the last thing I need is carbon all over the sump.

Finally, what's a good cheap brand of carbon to use purely to remove oxidised compounds (bromate?) coming from the skimmer?
I've read the carbon cleaning water/air from ozone doesn't need to be as good as the carbon used to absorb organics in reef tanks.

 

[WVNed]

I collected the air leaving the skimmer through the lid and made a housing out of PVC to hold carbon.
I did not put it in the cup. I reasoned that the air would simply pass over it that way.

It still skims at times when I am not looking and I have to empty it occasionally.

I used the cheapest carbon I could find since it was never in contact with the tank water.

I do not treat the water leaving the skimmer. Skimmers pump out a lot of water. Anything it will need to pass through will need to be large and have little restriction

I use an ORP controller and it turns the generator on and off. The skimmer just runs.