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.

src/app/chat/components/message-avatar.tsx

@@ -1,32 +1,61 @@
 "use client";
-import Image from "next/image";
 
-import { UserMessageAvatar } from "@/app/_components";
+import { CharacterAvatar, UserMessageAvatar } from "@/app/_components";
 import { useActiveCharacter } from "@/providers/character-provider";
 
 export interface MessageAvatarProps {
   isFromAI: boolean;
   userAvatarUrl?: string | null;
+  onClick?: () => void;
 }
 
 const AVATAR_CLASS_NAME =
-  "flex size-(--chat-avatar-size,43px) shrink-0 items-center justify-center overflow-hidden rounded-full border-2 border-(--color-avatar-border,#fbf3f5) bg-(--color-avatar-border,#fbf3f5) shadow-(--shadow-input-box,0_1px_2px_rgba(0,0,0,0.1))";
+  "size-(--chat-avatar-size,43px) ring-2 ring-(--color-avatar-border,#fbf3f5) shadow-(--shadow-input-box,0_1px_2px_rgba(0,0,0,0.1))";
 
-export function MessageAvatar({ isFromAI, userAvatarUrl }: MessageAvatarProps) {
+export function MessageAvatar({
+  isFromAI,
+  userAvatarUrl,
+  onClick,
+}: MessageAvatarProps) {
   const character = useActiveCharacter();
 
   if (isFromAI) {
-    return (
-      <div className={AVATAR_CLASS_NAME} aria-label="AI avatar">
-        <Image
+    if (onClick) {
+      return (
+        <CharacterAvatar
           src={character.assets.avatar}
           alt={character.displayName}
-          width={43}
-          height={43}
+          size="var(--chat-avatar-size, 43px)"
+          imageSize={43}
           priority
-          className="size-full object-cover"
+          className={AVATAR_CLASS_NAME}
+          actionLabel={`Open ${character.displayName}'s private room`}
+          analyticsKey="chat.open_private_room_from_avatar"
+          onClick={onClick}
         />
-      </div>
+      );
+    }
+
+    return (
+      <CharacterAvatar
+        src={character.assets.avatar}
+        alt={character.displayName}
+        size="var(--chat-avatar-size, 43px)"
+        imageSize={43}
+        priority
+        className={AVATAR_CLASS_NAME}
+      />
+    );
+  }
+
+  if (onClick) {
+    return (
+      <UserMessageAvatar
+        avatarUrl={userAvatarUrl}
+        actionLabel="Open profile"
+        analyticsKey="chat.open_profile_from_avatar"
+        onClick={onClick}
+      />
     );
   }