The context window is not on your system. It's on the server with the model. There may be some local prompt caching, of some sort, but you're not locally hosting the context unless you're also locally hosting the model.
Chat history is kept locally, generally you have to send the 'whole history' to the model 'each turn'.
Chat history is kept locally, generally you have to send the 'whole history' to the model 'each turn'.