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/game/Chat.svelte

@@ -56,7 +56,10 @@
     display: flex;
     flex-direction: column;
     gap: 10px;
+    /* dvh so the chat shrinks with an open keyboard, keeping the start of the
+       conversation on screen instead of pushed above the fold (vh fallback). */
     height: 56vh;
+    height: 56dvh;
   }
   .list {
     flex: 1;