Refactor and optimize: deduplicate, batch queries, fix bugs

- Remove unused DATABASE_PATH import from games.py
- Extract _new_game_context(db) helper to deduplicate GET/POST error path
- Extract _parse_hole_form() helper to deduplicate hole validation in create/edit course
- Batch N+1 player handicap and tee queries in create_game into two IN() queries
- Fix hole count import bug: use max() across all tees instead of first match
- Fix migration exception handling: re-raise unless "duplicate column name"
- Cache calcSkins() result in _skinsCache; invalidate on score/skip changes
- Collapse two identical Next buttons into one in new_game.html
- Fix stale skinDoubleBackNine checkbox: reset on mode change, use x-model

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

app/templates/games/new_game.html

@@ -163,7 +163,7 @@
             <div x-show="skinValue && holesMode === '18'">
               <label style="display:flex; align-items:center; gap:0.5rem; cursor:pointer;">
                 <input type="checkbox" name="skin_double_back_nine" value="1" style="width:auto;"
-                       @change="skinDoubleBackNine = $event.target.checked">
+                       x-model="skinDoubleBackNine">
                 <span style="font-weight:400; color:var(--text-muted);">Double value on back nine (holes 10–18)</span>
               </label>
             </div>
@@ -226,11 +226,11 @@
               <span>18 holes</span>
             </label>
             <label class="radio-card" style="flex:1;">
-              <input type="radio" name="_holes_mode" value="front9" @change="holesMode = 'front9'">
+              <input type="radio" name="_holes_mode" value="front9" @change="holesMode = 'front9'; skinDoubleBackNine = false">
               <span>Front 9</span>
             </label>
             <label class="radio-card" style="flex:1;">
-              <input type="radio" name="_holes_mode" value="back9" @change="holesMode = 'back9'">
+              <input type="radio" name="_holes_mode" value="back9" @change="holesMode = 'back9'; skinDoubleBackNine = false">
               <span>Back 9</span>
             </label>
           </div>
@@ -268,8 +268,7 @@
       </div>
 
       <div style="position:sticky; bottom:calc(var(--nav-height) + env(safe-area-inset-bottom)); background:var(--bg); padding:0.75rem 0; margin-top:0.5rem;">
-        <button type="button" x-show="hasTees()" class="btn btn-primary" @click="step = 2" :disabled="!holesMode">Next →</button>
-        <button type="button" x-show="!hasTees()" class="btn btn-primary" @click="goToConfirm()" :disabled="!holesMode">Next →</button>
+        <button type="button" class="btn btn-primary" @click="hasTees() ? step = 2 : goToConfirm()" :disabled="!holesMode">Next →</button>
       </div>
     </div>