Solved thanks to someone on Google+ 🙂
Your web host has forgotten to set up a tmp directory or to make it writable for you. If you forward the same error messages to them and mention the tmp directory they should resolve it for you.
The alternative is to override the default tmp directory by adding this line to you wp-config:
define( ‘WP_TEMP_DIR’, ABSPATH . ‘wp-content/tmp/’) ;
Then create that tmp folder in your wp-content directory and set permissions to 755. Really, though, it should be up to your host to correct.