Can you open the file app/code/Axis/Admin/controllers/Catalog/IndexController.php and add these lines:

Axis::message()->addError($this->_getParam('image'));
return $this->_helper->json->sendFailure();

right before the foreach loop (line ~247)

foreach ($jsonParams as $param) {...

Then try to save product again. You will recieve the error message. Post it here please.

the error is:

{\"ext-record-8\":{\"path\":\"/i/n/inel_logodna-2.jpeg\",\"sort_order\":50,\"is_thumbnail\":1,\"is_listing\":1,\"is_base\":1,\"remove\":0}}

Looks like you have magic_qutes option enabled in your server environment.
Disable it and try again to save the product.

it was disabled already by htaccess:

##################################

## Disable magic quotes

php_value magic_quotes_gpc off

Your server is probably running in CGI/FCGI mode, and this statement doesn't affect the environment.

The problem is that params recieved by server are being escaped by backslashes:

{\"ext-record-8\":{\"path\":\"/i/n/inel_logodna-2.jpeg\", ...}}

Param should not contain these backslashes:

{"ext-record-8":{"path":"/i/n/inel_logodna-2.jpeg", ...}}