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/account/profile_test.go

@@ -5,22 +5,29 @@ import (
 	"errors"
 	"strings"
 	"testing"
+	"time"
 
 	"github.com/google/uuid"
 )
 
 // TestUpdateProfileValidation checks that bad fields are rejected before any
-// database access, so a nil-backed Store is enough to exercise the guards.
+// database access, so a nil-backed Store is enough to exercise the guards. It also
+// confirms UpdateProfile wires the Stage 8 validators (name format, away window,
+// offset/IANA timezone), not just their unit tests in validate_test.go.
 func TestUpdateProfileValidation(t *testing.T) {
 	s := &Store{}
 	base := ProfileUpdate{DisplayName: "Kaya", PreferredLanguage: "en", TimeZone: "UTC"}
+	hm := func(h, m int) time.Time { return time.Date(0, 1, 1, h, m, 0, 0, time.UTC) }
 	tests := []struct {
 		name string
 		mut  func(p *ProfileUpdate)
 	}{
 		{"unknown language", func(p *ProfileUpdate) { p.PreferredLanguage = "fr" }},
 		{"invalid timezone", func(p *ProfileUpdate) { p.TimeZone = "Mars/Olympus" }},
+		{"bad offset timezone", func(p *ProfileUpdate) { p.TimeZone = "+15:00" }},
 		{"over-long name", func(p *ProfileUpdate) { p.DisplayName = strings.Repeat("x", maxDisplayName+1) }},
+		{"bad name layout", func(p *ProfileUpdate) { p.DisplayName = "Bad__Name" }},
+		{"away over 12h", func(p *ProfileUpdate) { p.AwayStart, p.AwayEnd = hm(8, 0), hm(21, 0) }},
 	}
 	for _, tc := range tests {
 		t.Run(tc.name, func(t *testing.T) {