Thanks for helping track these issues down.

Great Job!

Everything is solved for me

Should be fixed.

Changes have been made in Git for this ticket:

Bug #8948: Fix sent-mail mbox display when switching identities

http://git.horde.org/diff.php/imp/js/compose.js?rt=horde-git&r1=69049e265915aa1f6a005b6631cc1bae051989ae&r2=76b9d11e4426fbe58fc31e27ed3193423b32a4c7

Changes have been made in Git for this ticket:

Bug #8948: Typo

http://git.horde.org/diff.php/imp/lib/Prefs/Identity.php?rt=horde-git&r1=5b0969a7d3fad81ae04676dc8731a70205943ec5&r2=6088ce0ba2705b74abb190964fb40428aa912467

It seems a lot better:
- The prefs are stored as expected (no INBOX. prefix).
- The prefs are displayed as expected.

A last "minor" issue appears when I try to compose email:
- Before switching between identities, it displays: "Save a copy in sent-mail"
- When I switch between identities, it display: "Save a copy in 
INBOX.sent-mail"

But, the mail is saved in the correct folder.

Meanwhile, there is a small typo error over there:

--- a/imp/lib/Prefs/Identity.php
+++ b/imp/lib/Prefs/Identity.php
@@ -515,7 +515,7 @@ class Imp_Prefs_Identity extends Horde_Prefs_Identity
      public function setValue($key, $val, $identity = null)
      {
          if ($key == 'sent_mail_folder') {
-            $val = IMP::folderPrefs($val, false);
+            $val = IMP::folderPref($val, false);
          }
          return parent::setValue($key, $val, $identity);
      }

OK - what about this?

Changes have been made in Git for this ticket:

Bug #8948: Fix namespace stripping for pref values

http://git.horde.org/diff.php/imp/lib/IMP.php?rt=horde-git&r1=e6a2822fd71cf145a13b4dbd52ad13035b7d8f90&r2=66a79e1de90217ec0b55c0744e86cf047e946c2b
http://git.horde.org/diff.php/imp/lib/Imap.php?rt=horde-git&r1=9f73d1ff74206b8e5d162ba79e0415027b305106&r2=66a79e1de90217ec0b55c0744e86cf047e946c2b

Agree.

So, my problems are from the other side:

While debugging the source code, I found that my problem come from 
IMP::folderPref('some folder', true) function.

According to a cyrus IMAP backend, this function should add a 'INBOX.' 
prefix, if the folder is not INBOX, right? It's not the case...

Debugging of "IMP::folderPref('SPAM', true)":
folder = SPAM ; append = true;
def_ns = Array ( [name] => INBOX. [delimiter] => . [type] => personal 
[hidden] => )
empty_ns = Array ( [name] => [delimiter] => . [type] => shared [hidden] => )
GLOBALS['imp_imap']->getNamespace(folder) = Array ( [name] => 
[delimiter] => . [type] => shared [hidden] => )
enter this if block
skip this if block
skip this if block as ns is not null

So, folder value is not changed...

This breaks things for me.  And it makes no sense - IMP:folderPref() 
MUST be false when saving to the backend.  That's the whole point of 
IMP::folderPref().

Changes have been made in Git for this ticket:

Revert "Bug #8948: Fix sent-mail selection for non-empty namespaces"
This reverts commit 9d2152b7ddcc976ca796ae9801154181af3fc149.

http://git.horde.org/diff.php/imp/lib/Prefs/Ui.php?rt=horde-git&r1=c4a8bbfa852ae49c87acf4806c51b8c241af2f12&r2=055adb5538a560ebfedbf44555f5fcd64159b465

Fine.

I've updated to latest git today. But, the issue is still there (cyrus 
IMAP backend).
Thus, I've attached a patch which solves the issue for me.

Works for me, except for an issue with sent-mail folder which was just 
fixed.  Considering the entire options UI was rewritten in the last 
few weeks, you may have fallen victim to a transient issue.

Changes have been made in Git for this ticket:

Bug #8948: Fix sent-mail selection for non-empty namespaces

http://git.horde.org/diff.php/imp/lib/Prefs/Ui.php?rt=horde-git&r1=375ca9376b445764324b61a623a9a4b2c9fac567&r2=9d2152b7ddcc976ca796ae9801154181af3fc149

Fine.

So, IMP strip the "INBOX." prefix: Normal behaviour
But, it does not add it again to recognize Draft/Trash/SPAM/sent-mail folder:

When I modify "Server and Folder Information" options, It saved 
correct values in database backend, but, IMP do not display (ie. 
recognize) the corresponding value (see screenshot attached).

This is precisely the expected behavior.  Mailbox names are stored 
without their namespace on purpose.  See IMP::folderPref().

On my HORDE/IMP install, based on cyrus imap 2.2 + mysql backend,
the 'INBOX.' prefix of any folder is stripped in my prefs backend:

-------------------------------------------------------------------
: pref_uid : pref_scope : pref_name : pref_value
-------------------------------------------------------------------
: vink : imp : drafts_folder : Draft
: vink : imp : trash_folder : Trash
: vink : imp : spam_folder : SPAM
--------------------------------------------------------------------

When I add 'INBOX.' in front of these elements, these folders are 
correctly recognized in the horde/imp framework.

I've same issue with identites about sent-mail folder:
When composing e-mail, imp complains of not beeing able to create 
"sent-mail", but INBOX.sent-mail already exists. If I mannually edit 
the corresponding prefs in my backend (ie. rewrite sent-mail to 
INBOX.sent-mail), the issue is solved.