chore(e2e): add ESLint + Prettier; fix real findings; dedupe BASE

The Playwright suite had no linter and no formatter — only tsc. Add flat-config ESLint
(typescript-eslint, type-aware) and Prettier (2-space, matching the suite's style).

Rules keep the ones that catch real TEST bugs and drop the noise:
  - no-floating-promises KEPT — an un-awaited request/assertion can let a test end before it runs,
    passing vacuously. It caught one: the SSE reader loop in sse-listener is now explicitly `void`.
  - no-unused-vars KEPT — caught three dead bindings (an unused adminToken fixture arg, an unused
    `api` arg, an unused JPEG_MAGIC import), all removed.
  - no-explicit-any OFF — all test code; `any` is the honest type for an untyped res.json() body or
    a page.evaluate() return.
  - no-empty-pattern OFF — Playwright's dependency-free fixtures are `async ({}, use) => {}`.

Refactor: `const BASE = process.env.E2E_FRONTEND_URL ?? '...'` was redeclared verbatim in 23
files — extracted to helpers/env.ts and imported, so a port/scheme change is one edit not a sweep.

Then `prettier --write`. Verified: eslint clean, tsc clean, prettier clean, desktop suite 210
passed / 1 skipped. (One mobile spec flaked once under retries:0 — a pre-existing cross-test
reflow-timing vector from the flakiness audit, not this change: the each-key edit is stable across
16 isolated runs and a clean full mobile re-run.)

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

e2e/fixtures/api-client.ts

@@ -47,7 +47,9 @@ export class ApiClient {
   }
 
   // ── Auth ───────────────────────────────────────────────────────────────
-  async join(displayName: string): Promise<{ jwt: string; pin: string; user_id: string; is_new: boolean }> {
+  async join(
+    displayName: string
+  ): Promise<{ jwt: string; pin: string; user_id: string; is_new: boolean }> {
     const { body } = await this.request<any>('POST', '/join', {
       body: { display_name: displayName },
       expectedStatus: [201],
@@ -55,7 +57,11 @@ export class ApiClient {
     return body;
   }
 
-  async recover(displayName: string, pin: string, opts: { expectedStatus?: number | number[] } = {}) {
+  async recover(
+    displayName: string,
+    pin: string,
+    opts: { expectedStatus?: number | number[] } = {}
+  ) {
     return this.request<any>('POST', '/recover', {
       body: { display_name: displayName, pin },
       expectedStatus: opts.expectedStatus ?? [200],
@@ -91,7 +97,9 @@ export class ApiClient {
   }
 
   async getConfig(adminToken: string): Promise<Record<string, string>> {
-    const { body } = await this.request<Record<string, string>>('GET', '/admin/config', { token: adminToken });
+    const { body } = await this.request<Record<string, string>>('GET', '/admin/config', {
+      token: adminToken,
+    });
     return body;
   }
 
@@ -118,7 +126,11 @@ export class ApiClient {
     });
   }
 
-  async unbanUser(token: string, userId: string, opts: { expectedStatus?: number | number[] } = {}) {
+  async unbanUser(
+    token: string,
+    userId: string,
+    opts: { expectedStatus?: number | number[] } = {}
+  ) {
     return this.request<void>('POST', `/host/users/${userId}/unban`, {
       token,
       expectedStatus: opts.expectedStatus ?? [200, 204],