Stage 8: regression tests for the review-round refinements

Lock the polish behaviours so a future edit surfaces as a failing test:
- backend: UpdateProfile now rejects a bad name layout, an away window over 12h, and
  a malformed offset timezone (confirming it wires the Stage 8 validators); a new
  integration test accepts and resolves a "+03:00" offset timezone.
- e2e (mock): the lobby notification badge count, the play-with-friends required
  game type + invitation send, the in-game add-to-friends flipping to a disabled
  "request sent", the profile-edit invalid-name Save guard, and the chat send/nudge
  icon buttons.

backend/internal/inttest/email_test.go

@@ -7,6 +7,7 @@ import (
 	"errors"
 	"regexp"
 	"testing"
+	"time"
 
 	"github.com/google/uuid"
 
@@ -157,3 +158,26 @@ func TestUpdateProfilePersists(t *testing.T) {
 		t.Errorf("profile did not persist: %+v", reloaded)
 	}
 }
+
+// TestUpdateProfileOffsetTimezone checks the Stage 8 UTC-offset timezone: it is
+// accepted, persisted verbatim, and resolved to the right fixed offset.
+func TestUpdateProfileOffsetTimezone(t *testing.T) {
+	ctx := context.Background()
+	store := account.NewStore(testDB)
+	acc := provisionAccount(t)
+
+	updated, err := store.UpdateProfile(ctx, acc, account.ProfileUpdate{
+		DisplayName:       "Kaya",
+		PreferredLanguage: "en",
+		TimeZone:          "+03:00",
+	})
+	if err != nil {
+		t.Fatalf("update with offset timezone: %v", err)
+	}
+	if updated.TimeZone != "+03:00" {
+		t.Fatalf("timezone = %q, want +03:00", updated.TimeZone)
+	}
+	if _, off := time.Date(2024, 1, 1, 12, 0, 0, 0, account.ResolveZone(updated.TimeZone)).Zone(); off != 3*3600 {
+		t.Fatalf("ResolveZone offset = %d, want 10800", off)
+	}
+}