I am entertaining the idea of making a spicy (apple cidar vinegar, sriracha,etc) cole slaw and am looking to temper the heat a bit with cucumber. Is the best option to slice in long strips? Is so, should I keep, discard, or otherwise incorporate the skin and the seed pulp?
If you want to match the texture of the shredded cabbage you could use a peeler to get thin, long strips of the cucumber. If you want to contrast the texture you could cut it into chunks or slices.
I would discard the seed pulp to cut down on the amount of water that will fall to the bottom of the bowl.
The skin is completely personal preference. If it's thin and tender I like to leave at least strips of it on for color contrast. Otherwise I get rid of it to avoid the tough, chewiness it can bring.
I would try using peeled cucumbers, because of the texture issues. You can grate them into a sieve, and then put some weight on them for half an hour or so. This will help drain some of the excess liquid.
This is the same preparation of the cucumbers that is used for cucumber raita, or tzaziki.