For both issues it is because your file permissions are wrong.
I am going to make two assumptions for the sake of example, you will have to change them where they differ on your installation. They are that your nginx server process is installed and running as a user called 'nginx' and you have a regular account that you are using for adding files to the server called 'webuser'.
First you need to set the permissions on your magento web server directory so that you 'webuser' are the owner of the files and 'nginx' as the group. To do this, as a user with root privileges issue the command...
chown -R webuser:nginx /path
where /path = the root path for your magento installation.
Next change the permissions as follows ...
find . -type d -exec chmod 770 {} \; && find . -type f -exec chmod 660 {} \; && chmod u+x bin/magento
Now when you run the cron make sure you are doing this as the owner of the file system, in this example called 'webuser'.
Do NOT run it as root as the files created will have root as the owner.
You may have to repeat the application of permissions after you have run the cron scripts to ensure all files and directories created by the scripts have the correct permissions.
