feat(profile)!: replace sidebar and add avatar navigation

Rename the global Sidebar route, UI, assets, analytics, and payment return context to Profile. Add accessible message-avatar navigation and preserve the source character across auth and logout flows.

BREAKING CHANGE: /sidebar has been removed; use /profile instead.

e2e/fixtures/helpers/auth.ts

@@ -11,7 +11,7 @@ export const splashStartChatButtonName = "Start Chatting";
 export const defaultCharacterSlug = "elio";
 export const defaultCharacterSplashPath = `/characters/${defaultCharacterSlug}/splash`;
 export const defaultCharacterChatPath = `/characters/${defaultCharacterSlug}/chat`;
-export const defaultCharacterSidebarPath = `/sidebar?returnTo=%2Fcharacters%2F${defaultCharacterSlug}%2Fchat`;
+export const defaultCharacterProfilePath = `/profile?returnTo=%2Fcharacters%2F${defaultCharacterSlug}%2Fchat`;
 export const defaultCharacterChatUrl = new RegExp(
   `/characters/${defaultCharacterSlug}/chat(?:\\?.*)?$`,
 );

e2e/fixtures/helpers/chat.ts

@@ -27,7 +27,7 @@ export async function enterChatFromSplash(page: Page, options: { expectedUrl?: R
   await expect(startChatButton).toBeVisible({ timeout });
   await expect(startChatButton).toBeEnabled();
   for (let attempt = 0; attempt < 3; attempt += 1) {
-    await startChatButton.click();
+    await startChatButton.click({ timeout: 3_000 }).catch(() => {});
     try { await expect(page).toHaveURL(expectedUrl, { timeout: 3_000 }); return; } catch { await page.waitForTimeout(500); }
   }
   await expect(page).toHaveURL(expectedUrl, { timeout });