Will bitter cucumbers ruin gazpacho

The biggest thing you can do is peel it. If a cucumber's bitter, it's much more in the peel than in the flesh. It's also usually more bitter at the stem end, so you might want to start from the other end, and possibly ultimately discard an inch from the stem end if it really is too bitter for you. Unless you're unusually sensitive to the bitterness, that ought to cover it.

Bitterness is also fairly variable from cucumber to cucumber, so it's possible that you could shop for less bitter ones - maybe a different variety - or buy them only during more favorable seasons.

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.