If found it quiet easy to change the default behavior of the "Post Reply" form to autoselect 'Feedback'. This is a HTML only change and does not affect the rest of the app logic.

diff --git a/include/staff/ticket-view.inc.php b/include/staff/ticket-view.inc.php
index 47e7728e..eabb599e 100644
--- a/include/staff/ticket-view.inc.php
+++ b/include/staff/ticket-view.inc.php
@@ -1106,12 +1106,13 @@ if ($errors['err'] && isset($_POST['a'])) {
                         foreach (TicketStatusList::getStatuses(
                                     array('states' => $states)) as $s) {
                             if (!$s->isEnabled()) continue;
-                            $selected = $statusId == $s->getId();
+                            $current = ($statusId == $s->getId());
+                            $selected = ($s->getName() == 'Feedback');
                             echo sprintf('<option value="%d" %s>%s%s</option>',
                                     $s->getId(),
                                     $selected ? 'selected="selected"' : '',
                                     __($s->getName()),
-                                    $selected ? (' ('.__('current').')') : ''
+                                    $current ? (' ('.__('current').')') : ''
                                     );
                         }
                         ?>

Then I tried to edit also class.ticket.php to change back the ticket status to 'InProgress' when a user replies,
but this didn't work. It looks like the onMessage handler is not called in this case, so I still lack some understanding of the code. The code is pretty easy to read, actually, but here I have need to investigate further. Maybe you have a tip.

diff --git a/include/class.ticket.php b/include/class.ticket.php
index 593df7d8..aba3b4fa 100644
--- a/include/class.ticket.php
+++ b/include/class.ticket.php
@@ -228,6 +228,10 @@ implements RestrictedAccess, Threadable, Searchable {
          return $this->hasState('closed');
     }
 
+    function isFeedback() {
+         return $this->hasState('Feedback');
+    }
+
     function isCloseable() {
         global $cfg;
 
@@ -1445,6 +1449,13 @@ implements RestrictedAccess, Threadable, Searchable {
         return $status ? $this->setStatus($status) : false;
     }
 
+    function changeToInProgress() {
+        if (!$this->isFeedback())
+            return false;
+
+        return $this->setStatus('In Progress');
+    }
+
     function onNewTicket($message, $autorespond=true, $alertstaff=true) {
         global $cfg;
 
@@ -1694,6 +1705,12 @@ implements RestrictedAccess, Threadable, Searchable {
         $this->save();
 
 
+        if ($this->isFeedback()) {
+            $this->changeToInProgress();
+            $this->save();
+            return;
+        }
+
         // Reopen if closed AND reopenable
         // We're also checking autorespond flag because we don't want to
         // reopen closed tickets on auto-reply from end user. This is not to
--