refactor: split App.vue into composables and layout components

Extract 11 composables and 5 layout components from App.vue, reducing it
from 1921 to 327 lines. Unify dialog overlay style across DialogContent
and DialogScrollContent for consistent backdrop appearance.

src/composables/useDatabaseOptions.ts

@@ -0,0 +1,40 @@
+import { ref } from "vue";
+import { useConnectionStore } from "@/stores/connectionStore";
+import * as api from "@/lib/api";
+
+export function useDatabaseOptions() {
+  const connectionStore = useConnectionStore();
+
+  const databaseOptions = ref<Record<string, string[]>>({});
+  const loadingDatabaseOptions = ref<Record<string, boolean>>({});
+
+  async function loadDatabaseOptions(connectionId: string) {
+    const connection = connectionStore.getConfig(connectionId);
+    if (!connection || loadingDatabaseOptions.value[connectionId]) return;
+
+    loadingDatabaseOptions.value[connectionId] = true;
+    try {
+      await connectionStore.ensureConnected(connectionId);
+      if (connection.db_type === "redis") {
+        const dbs = await api.redisListDatabases(connectionId);
+        databaseOptions.value[connectionId] = dbs.map(String);
+      } else if (connection.db_type === "mongodb") {
+        databaseOptions.value[connectionId] = await api.mongoListDatabases(connectionId);
+      } else {
+        const dbs = await api.listDatabases(connectionId);
+        databaseOptions.value[connectionId] = dbs.map((db) => db.name);
+      }
+    } finally {
+      loadingDatabaseOptions.value[connectionId] = false;
+    }
+  }
+
+  async function getDatabaseOptions(connectionId: string): Promise<string[]> {
+    if (!databaseOptions.value[connectionId]) {
+      await loadDatabaseOptions(connectionId);
+    }
+    return databaseOptions.value[connectionId] ?? [];
+  }
+
+  return { databaseOptions, loadingDatabaseOptions, loadDatabaseOptions, getDatabaseOptions };
+}