feat(FN-2232): check mailbox on every heartbeat run

- Fetch unread inbox messages whenever messageStore is available, not only wake-on-message triggers
- Include pending messages in execution prompts for task and no-task heartbeat runs across timer, assignment, and on-demand sources
- Extend heartbeat messaging tests to cover run-wide mailbox prefetch and read-marking behavior
- Document heartbeat mailbox checking semantics and messageResponseMode behavior in agents docs
- Replace Header mailbox badge class with existing header-badge style token

packages/dashboard/app/components/Header.tsx

@@ -928,7 +928,7 @@ export function Header({
           >
             <Mail size={16} />
             {mailboxUnreadCount > 0 && (
-              <span className="btn-icon-indicator" data-testid="header-mailbox-badge">
+              <span className="header-badge" data-testid="header-mailbox-badge">
                 {mailboxUnreadCount > 9 ? "9+" : mailboxUnreadCount}
               </span>
             )}