[FEATURE] Add a "New chat" button / shortcut to quickly start a fresh session in the VS Code chat panel

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

In the Claude Code VS Code extension chat panel, there's no button to copy an assistant response. Copying requires manually selecting text with the mouse, which is slow and loses or corrupts markdown formatting. The Claude web and desktop apps both have a per-message copy button — the extension doesn't. This was previously raised in #16268, which is now closed and locked, so I'm filing a fresh request as suggested by the auto-lock bot.

Proposed Solution

Add a "Copy" button (icon) that appears on hover, or is always visible, at the top of each assistant response in the chat panel — identical to the Claude web app. Clicking it copies the full response with markdown formatting preserved.

Alternative Solutions

Manual text selection + Ctrl+C, which is error-prone and breaks formatting on multi-line content.

Priority

Nice to have

Feature Category

IDE / VS Code extension.