feat: add search APIs and telegram-like chats sidebar flow

- implement chat query filtering and message search endpoints

- add db indexes for search fields

- activate chats search input in web

- replace inline create panel with floating TG-style action menu

web/src/store/chatStore.ts

@@ -7,7 +7,7 @@ interface ChatState {
   activeChatId: number | null;
   messagesByChat: Record<number, Message[]>;
   typingByChat: Record<number, number[]>;
-  loadChats: () => Promise<void>;
+  loadChats: (query?: string) => Promise<void>;
   setActiveChatId: (chatId: number | null) => void;
   loadMessages: (chatId: number) => Promise<void>;
   prependMessage: (chatId: number, message: Message) => void;
@@ -29,9 +29,11 @@ export const useChatStore = create<ChatState>((set, get) => ({
   activeChatId: null,
   messagesByChat: {},
   typingByChat: {},
-  loadChats: async () => {
-    const chats = await getChats();
-    set({ chats, activeChatId: chats[0]?.id ?? null });
+  loadChats: async (query) => {
+    const chats = await getChats(query);
+    const currentActive = get().activeChatId;
+    const nextActive = chats.some((chat) => chat.id === currentActive) ? currentActive : (chats[0]?.id ?? null);
+    set({ chats, activeChatId: nextActive });
   },
   setActiveChatId: (chatId) => set({ activeChatId: chatId }),
   loadMessages: async (chatId) => {