" Some of my red chili peppers bloomed just fine but now, many of the newer chilis (and the leaves for that matter) have dark, black spots. Are my plants diseased? What’s happening and how can I fix it?"
You definitely have something damaging your plants. It sure looks like thrips damage and you should consider very carefully if you really want to save these plants. They will migrate to other house plants and it is super difficult to get rid of them. Also, they are difficult to spot as they are experts in hiding and super tiny. There are some suggestions about bug problems here: https://support.clickandgrow.com/hc/en-us/articles/115009431968-General-plant-health-care-tips-and-tricks
Next time, if multiple seeds germinate in each pod in case of chili, leave one seedling to grow per plant pod to ensure enough space and nutrients for the plant to grow.
You can find more useful plant care tips in C&G webstore, under products.