refactor(ui): extract shared integration grid logic and deduplicate

Both IntegrationMultiSelectGrid and IntegrationSelectGrid had
duplicated logic for building widget-by-integration maps, category
labels, sorting, and filtering. Extract shared code into
integration-grid-shared.ts using objectEntries and IntegrationKind
typing. Use i18n keys for category badges and widget counts instead
of hardcoded strings. Remove useEffect for onSelectionChange in
multi-select grid — call it directly in the toggle callback.

apps/nextjs/src/components/integration/integration-grid-shared.ts

@@ -0,0 +1,84 @@
+import { objectEntries } from "@homarr/common";
+import type { IntegrationKind, WidgetKind } from "@homarr/definitions";
+import { featuredIntegrations, getIntegrationName, integrationDefs, integrationKinds } from "@homarr/definitions";
+import { widgetImports } from "@homarr/widgets";
+
+export const CARD_HEIGHT = 180;
+
+const buildWidgetsByIntegration = () => {
+  const map = Object.fromEntries(integrationKinds.map((kind) => [kind, [] as WidgetKind[]])) as Record<
+    IntegrationKind,
+    WidgetKind[]
+  >;
+
+  for (const [widgetKind, widget] of objectEntries(widgetImports)) {
+    const supported =
+      "supportedIntegrations" in widget.definition
+        ? (widget.definition.supportedIntegrations as IntegrationKind[])
+        : [];
+    for (const kind of supported) {
+      if (kind in map) {
+        map[kind].push(widgetKind);
+      }
+    }
+  }
+  return map;
+};
+
+export const widgetsByIntegration = buildWidgetsByIntegration();
+
+export const categoryTranslationKeys: Record<string, string> = {
+  dnsHole: "integration.category.dnsHole",
+  mediaService: "integration.category.mediaService",
+  calendar: "integration.category.calendar",
+  mediaSearch: "integration.category.mediaSearch",
+  mediaRelease: "integration.category.mediaRelease",
+  mediaRequest: "integration.category.mediaRequest",
+  downloadClient: "integration.category.downloadClient",
+  usenet: "integration.category.usenet",
+  torrent: "integration.category.torrent",
+  miscellaneous: "integration.category.miscellaneous",
+  smartHomeServer: "integration.category.smartHomeServer",
+  indexerManager: "integration.category.indexerManager",
+  healthMonitoring: "integration.category.healthMonitoring",
+  search: "integration.category.search",
+  mediaTranscoding: "integration.category.mediaTranscoding",
+  networkController: "integration.category.networkController",
+  releasesProvider: "integration.category.releasesProvider",
+  notifications: "integration.category.notifications",
+  firewall: "integration.category.firewall",
+  timetable: "integration.category.timetable",
+  photoService: "integration.category.photoService",
+  notes: "integration.category.notes",
+  mediaMonitoring: "integration.category.mediaMonitoring",
+  speedtest: "integration.category.speedtest",
+  analytics: "integration.category.analytics",
+};
+
+export interface IntegrationGridItem {
+  kind: IntegrationKind;
+  name: string;
+  categories: string[];
+  widgets: WidgetKind[];
+}
+
+export const buildSortedIntegrations = (enableMockIntegration: boolean): IntegrationGridItem[] =>
+  integrationKinds
+    .filter((kind) => enableMockIntegration || kind !== "mock")
+    .map((kind) => ({
+      kind,
+      name: getIntegrationName(kind),
+      categories: [...new Set(integrationDefs[kind].category.flat())] as string[],
+      widgets: widgetsByIntegration[kind],
+    }))
+    .sort((left, right) => {
+      const leftIdx = featuredIntegrations.indexOf(left.kind);
+      const rightIdx = featuredIntegrations.indexOf(right.kind);
+      if (leftIdx !== -1 && rightIdx !== -1) return leftIdx - rightIdx;
+      if (leftIdx !== -1) return -1;
+      if (rightIdx !== -1) return 1;
+      return right.widgets.length - left.widgets.length || left.name.localeCompare(right.name);
+    });
+
+export const filterIntegrations = (items: IntegrationGridItem[], search: string) =>
+  items.filter((i) => i.name.toLowerCase().includes(search.toLowerCase().trim()));