White stuff on the top of the apple cider vinegar

There shouldn't be any significant difference between an established homemade sourdough culture and one that's seeded from something you bought (I'm assuming like the culture that King Arthur Flour Co sells online). In fact, no matter where you bought your starter culture, over time the local bacterial flora would crowd out the bacteria that was in the culture you bought. But, I'm sure the mold infection had nothing to do with rogue bacteria. (Bad bacteria can spoil a starter too, but that would be a different set of symptoms from the green carpet on top.) It could be that your starter didn't get well established, but it doesn't sound like that was the problem if you successfully made some bread from it.

In a sourdough culture, the acid produced by the bacteria and alcohol produced by the yeast make an environment that's somewhat resitant to bad bacteria (like salmonella) or mold. But the culture needs to be fairly active to maintain its resistance. After two weeks in a warm fridge, your yeast would be dormant and the bacteria would probably have run rampant for a while (after the yeast went dormant) and then started to also die off.

In a warm fridge you could maybe go a week without feeding the culture. At room temperature it's one day optimally and two days max. The best way to avoid infection is to have two small covered containers about 1½ to 2 cups in size. To make a new generation of your starter, begin with a clean container, add equal parts by weight of the previous generation of starter, flour and water. So for example, 25 grams of starter, plus 25 grams of flour and 25 of water. Mix it up and put it in the fridge. In one week, get the other container, clean it out, and do the same thing over again.

I always keep the old generation in its container as a backup (in case of a mold or other infection). You just need to keep track of which container has the fresher starter, so you can clean out the old one when building the next gen. I did this for more than 3 years with no infections of any kind.

Don't eat it! The soup or the chicken.

Chicken meat may contain bacteria (inside the meat, not just surface) and can grow. Even if you've kept in the fridge for the weeks, it can still carry enough bad bacteria to seriously harm you. From what I know, the meat and fat structure of chicken doesn't lend it to curing in open air and long term.

If you are in fact trying to make a beef-jerky or cured sausage style and then cook in the soup to give the soup flavour and maintain decent flavour in the chicken, then you should be doing it differently.

You can dry cure the chicken in a dehydrator set to temperatures above 140°F as in this recipe.

For the soup: you'll need a soup base that is nearly as salty as the chicken (say %6 solution) so the salt doesn't leave the chicken and try to establish a new equilibrium between the saltiness of the chicken and soup.

This may mean that you might have to hold a portion of your soup back to add after the chicken is pulled out so it becomes less salty, but maintain the chicken flavour.