At least I could not get claude projects working, after it worked perfectly about 2 months ago.
Because Anthropic introduced a chatbot as only line of support, help is not on is way.
In many cases not. E.g. for caching with python, diskcache is a good choice.
For small amounts of data, a JSON file does the job (you pointed to JSONL as an option).
But for larger collections, that should be searchable/processable, postgres is a good choice.
Memory of course, as you wrote, also seems reasonable in many cases.