No, I cannot access themes from W.org through the Dashboard. I have to download and upload. I can manually install plugins through the Dashboard. It’s just updates of plugins that don’t work. I even tried a clean install of WP 4.0.1 under another user and with zero plugins active I got the notification to upgrade WP and the themes. It just did not do anything with Askimet or one old plugin (WP-Optimize) that was installed but not active. Even after upgrading to 4.5.3 there were no alerts to upgrade the plugins.
I have been working with the host to troubleshoot this as well. I have root access on my VPS which is why I was able to turn off the firewall. I also upgraded to php7 and Apache 2.4.