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Hm... Now that I am looking at a good sized fruit, I can see something I didn't notice before. That actually looks like an Armenian Cucumber! If this is, in fact, the case... That thing is at the perfect size to be picked! Regarding the rotting end... This is a common symptom of magnesium, or calcium deficiency. You should apply dolomitic lime to the area to restore these minerals. It sounds like you are doing a good job with your crops. Enjoy the climate, because you are in the perfect area!

They are very closely related, and will have similar care. It would be the same for the cuttings. It is like the zucchini in that cuttings are very hard to root. Those seeds will likely come back the same though. Those are less likely to breed out, therefore the seeds usually remain stable. All you will have to do in this case, and ask for a few of those seeds from the fruit.

As for how to use them... They are used just like any other cucumber. You may treat them as such, although they are quite different then their English counterpart.

This is just plant sap. Try letting them ripen just a little longer before picking them and this will reduce the sap some. As for removing it. I would try either peroxide and baking soda, or white vinegar and baking soda. These are what I use for tougher stains. You will want to do it as soon as you see that it is on something, though. The stain will quickly set if left for too long.

Lack of sunlight is the most common reason for leggy seedlings.
This article will help you.

https://www.gardeningknowhow.com/garden-how-to/propagation/seeds/leggy-seedling.htm

There are some things that you can do. This article will help: https://www.gardeningknowhow.com/edible/vegetables/squash/squash-pests-identifying-and-preventing-squash-vine-borer.htm