chore(lint): biome cleanup across services — format, sort imports, drop unused

- discord-gateway: 74 lint errors -> 0 (format, import sorting, unused
  imports/vars, dead breath var)
- backend: format + sort imports (11 warnings left: noExplicitAny)
- frontend: remove unused imports, drop dead breathing var, fix
  useExhaustiveDependencies (scroll keyed on messages), a11y biome-ignore
  for drag surface + stopPropagation container (mouse-only gestures)
- remaining warnings are false positives: index keys on static lists,
  <img> in static export (next/image unsupported), noExplicitAny

tsc --noEmit clean on all 3 services; vitest green (60+36).

services/discord-gateway/src/modules/event-broadcaster/eventBroadcaster.ts

@@ -1,6 +1,9 @@
-import type { AttachmentRecord, MessageRecord } from "../../shared/index.js";
-import { type CustomLogger, createChildLogger } from "../../shared/logger/index.js";
 import Redis from "ioredis";
+import type { AttachmentRecord, MessageRecord } from "../../shared/index.js";
+import {
+  type CustomLogger,
+  createChildLogger,
+} from "../../shared/logger/index.js";
 import { type DiscordGatewayEvent, EventChannels } from "./eventTypes.js";
 
 export class RedisEventPublisher {