I have several fruits lurking under the foliage of my plant, but they were mottled dark and light green last time I checked and the pale bits need to go white before they are ripe.
Storage is very good. I was given one fruit, but have been told a year is perfectly achievable in a shed. The Chinese simmer them for 2 hours plus to make a fake sharks fin soup - hence the name. They flesh breaks down into shreds and is very good at absorbing flavours, plus the broth it is cooked in, so I reckon it would be fairly gelatinous given that treatment.
It is grown quite widely in different countries and known by different names, so try searching Malabar Gourd, Fig Leaf Gourd or Chilacayote. It does make a kind of vegetable noodle in soup, but there are also recipes for cooking it with meat to jam-like preserves. You can eat the seeds roasted or make a kind of sweet like peanut brittle with them.
It has a slightly sweet mild flavour but as said, it very good at soaking up whatever it is cooked with. I ate my donated fruit as soup noodles with homemade chicken stock and plot greens wilted into it, which was lovely, but now I have more I will be experimenting