Can multiple separate conversations exist within a single Project?

Yes. A Project is a workspace with its own chat history, not a single long thread, so you can start as many separate conversations inside it as the work needs — one per bug, one per chapter, one per experiment — and each keeps its own transcript.

What they share is the Project's knowledge base and its instructions. Files, documents and code you upload sit at the Project level, and Claude draws on them in every chat you start there, along with any project instructions you have set. That is why a new conversation in a Project does not begin from nothing: the background is already attached, without you pasting it again.

The sharing runs one way, and it is worth knowing which way. Project knowledge reaches every chat, but the chats stay separate transcripts, so what you worked out in one is not automatically part of another. If something that surfaced in a conversation matters to the whole effort, add it to the Project's knowledge yourself.