Stage 8 polish: iPhone refinements (keyboard, native pickers, compact invite)

Second owner-review pass (iPhone simulator):
- Chat (and the modal) are sized in dvh so they shrink above the software keyboard,
  keeping the start of the conversation on screen instead of pushed off the top.
- The profile away window returns to a native <input type="time" step="600"> (the iOS
  wheel with 10-minute steps) instead of separate dropdowns; the timezone stays a
  native offset <select>.
- A finished game reserves the rack's height (min-height) so the footer no longer
  collapses when the final rack is empty — no layout jump versus an active game.
- New-game "play with friends" is made compact: a searchable, bounded-scroll friend
  list, the game-type / move-time / hints controls as native selects in one row
  (labels above), and Send invitation pinned at the bottom — it scales to many friends.

ui/src/lib/i18n/en.ts

@@ -196,6 +196,8 @@ export const en = {
   'new.auto': 'Quick match',
   'new.withFriends': 'Play with friends',
   'new.pickFriends': 'Choose who to invite',
+  'new.searchFriends': 'Search friends',
+  'new.gameType': 'Game type',
   'new.invite': 'Send invitation',
   'new.moveTime': 'Move time',
   'new.hintsPerPlayer': 'Hints per player',

ui/src/lib/i18n/ru.ts

@@ -197,6 +197,8 @@ export const ru: Record<MessageKey, string> = {
   'new.auto': 'Быстрая игра',
   'new.withFriends': 'Игра с друзьями',
   'new.pickFriends': 'Кого пригласить',
+  'new.searchFriends': 'Поиск друзей',
+  'new.gameType': 'Тип игры',
   'new.invite': 'Отправить приглашение',
   'new.moveTime': 'Время на ход',
   'new.hintsPerPlayer': 'Подсказок на игрока',