Please help!

When I want to save K2 articles i always get an
"error 500" or "Internal Server Error"

Things I´ve done until now:

There was a problem with folder premissions for /log and /tmp folder.
I solved it - in configuration.php - i added two dotts before slash
public $log_path = '../logs/';
public $tmp_path = '../tmp/';

And via FTP program i set the premissions to 777...
In Joomla System Info the two folders are shown with the writeable tag.

In some forums i red about a "k2 smart search" plugin issue in context with the error 500.
I´ve deactivated the plugin. Then the "Internal Server Error" disapeared.
And in the backend now I can edit and save articles normally.

But in the frontend editor I can open the articles but i can`t save them.
There always is following error message:

JFolder::create: Der Pfad ist nicht in den „open_basedir“-Pfaden!
Call stack
# Function Location
1 JSite->dispatch() JROOT/index.php:43
2 JComponentHelper::renderComponent() JROOT/includes/application.php:197
3 JComponentHelper::executeComponent() JROOT/libraries/joomla/application/component/helper.php:351
4 require_once() JROOT/libraries/joomla/application/component/helper.php:383
5 JController->execute() JROOT/components/com_k2/k2.php:62
6 K2ControllerItem->save() JROOT/libraries/joomla/application/component/controller.php:761
7 K2ModelItem->save() JROOT/components/com_k2/controllers/item.php:164
8 JPluginHelper::importPlugin() JROOT/administrator/components/com_k2/models/item.php:186
9 JPluginHelper::_import() JROOT/libraries/joomla/plugin/helper.php:125
10 plgFinderCategories->__construct() JROOT/libraries/joomla/plugin/helper.php:194
11 FinderIndexerAdapter->__construct() JROOT/plugins/finder/categories/categories.php:85
12 FinderIndexerAdapter->getTypeId() JROOT/administrator/components/com_finder/helpers/indexer/adapter.php:135
13 JLog::add() JROOT/administrator/components/com_finder/helpers/indexer/adapter.php:765
14 JLog->addLogEntry() JROOT/libraries/joomla/log/log.php:171
15 JLoggerFormattedText->addEntry() JROOT/libraries/joomla/log/log.php:382
16 JLoggerFormattedText->initFile() JROOT/libraries/joomla/log/loggers/formattedtext.php:137
17 JFolder::create() JROOT/libraries/joomla/log/loggers/formattedtext.php:237
18 JFolder::create() JROOT/libraries/joomla/filesystem/folder.php:188
19 JError::raiseWarning() JROOT/libraries/joomla/filesystem/folder.php:249
20 JError::raise() JROOT/libraries/joomla/error/error.php:276

Call stack
# Function Location
1 JSite->dispatch() JROOT/index.php:43
2 JError::raiseError() JROOT/includes/application.php:208
3 JError::raise() JROOT/libraries/joomla/error/error.php:251

Here is the error message when I activate the smart search plugin:
k2 Internal Server Error The server encountered an internal error or misconfiguration and was unable to complete your request. Please contact the server administrator, root@localhost and inform them of the time the error occurred, and anything you might have done that may have caused the error. More information about this error may be available in the server error log.

I find many posts about "finder_tokens Tabelle is full" They describe a workearround to increase the memory for tables in com -> smartsearch -> options => but no success in my case.

versions:
Joomla! 2.5.14
K2 v2.6.7

Please help - thank you

I can see several issues here.
The most important is that you are running outdated versions both of Joomla! and K2 .
Please update to J! 2.5.27 and K2 2.6.8 or K2 2.7 DEV.

Now the finder plugin is misbehaving. You can safely deactivate the smartsearch (core Joomla! stuff) if you are not using them.

You also need to check if the path to your /tmp and /log files is correct.

Finally you need to install an extension like Admin Tools to fix your file/ folder permissions.
Manually setting 777 permissions can raise certain issues.
If the extension cannot do this, you need to contact your hosting provider about this.