@vladax, There two possible issues:
1) Somehow the plugin files on your install is corrupt. So I would delete the plugin completely from your site then reinstall and try it.
2) If the above still doesn't fix it then you have some issue(s) in the server configuration. Or our plugin is not compatible with your server config. We test the plugin on the most widely used server setup and we know it doesn't have any of the issues you mentioned (I just did another test now).
If you are not using one of the WordPress's recommended hosting then this sort of server configuration related issues can happen:
I would recommend you to give the following plugin a go... maybe it will work better for your server setup: