What they mean when they say 'cached' is that it is loaded into the GPU memory on anthropic servers.
You already have the data on your own machine, and that 'upload and restore' process is exactly what is happening when you restart an idle session. The issue is that it takes time, and it counts as token usage because you have to send the data for the GPU to load, and that data is the 'tokens'.
Wrong on both counts. The kv-cache is likely to be offloaded to RAM or disk. What you have locally is just the log of messages. The kv-cache is the internal LLM state after having processed these messages, and it is a lot bigger.