Cannot reproduce either.

Please provide the FULL backtrace if you desire help in tracking this down.

Problem seems to be related to the fact that every time I log in to 
Horde installation a new address book,  a new task list and a new 
calendar is automatically created. It is as if the horde or Imp is 
unable to see or find that I already have a address books or a task 
lists or calendars and creates a new ones with default settings.

What might cause this automatic creation of address books or task 
lists and calendars?

Trying to edit newly created address book or existing address books 
results in blank page. No errors in logs.

Trying to access calendars in kronolith results in error:  HORDE: 
share parameter is not a Horde_Share_Object [pid 28560 on line 1600 of 
"/horde/framework/Core/lib/Horde/Registry.php"]

Snippet of backtrace:
                     [file] => /horde/framework/Core/lib/Horde/Registry.php
                     [line] => 1561
                     [function] => _pushAppError
                     [class] => Horde_Registry
                     [type] => ->
                     [args] => Array
                         (
                             [0] => InvalidArgumentException Object
                                 (
                                     [message:protected] => share 
parameter is not a Horde_Share_Object
                                     [string:Exception:private] =>
                                     [code:protected] => 0
                                     [file:protected] => 
/horde/kronolith/lib/Calendar/Internal.php
                                     [line:protected] => 37
                                     [trace:Exception:private] => Array

You need to track this down yourself then.

I tried to do this:
1. Do git pull
2. Do install_dev
3. Restart web server
4. Check log for errors: none
4. Login in to Horde (authentication is done with Imp application)
5. Initial application opens: Imp
6. Trying to do logout from Imp

This results in a blank page and Apache error log has an entry:
PHP Fatal error:  Call to a member function get() on a non-object in 
/horde/turba/lib/Turba.php on line 491, referer: 
https://example.com/horde/imp/dynamic.php?page=mailbox

No others errors in a Apache log file.

Cannot reproduce. There are probably earlier errors in one of the 
logs, please check.

I tried to do these things, and they result blank page and PHP fatal errors:
* Trying to access administration / configuration page
* Trying to logout from Horde

Doing what?

Error:
PHP Fatal error:  Call to a member function get() on a non-object in 
/hordenx/turba/lib/Turba.php on line 488

Line 488 seems to be:
$personal |= ($share->get('owner') == $auth_user);

Similar problem also seems to exist in Mnemo:
PHP Fatal error:  Call to a member function get() on a non-object in 
/hordenx/mnemo/lib/Mnemo.php on line 260

Line 260 seems to be:
$label = $notepad->get('name');