Question: My Pi camera will not work with the Pi zero using the same python code I use on my Pi 3 or Pi 2
Is there something I am missing?

Answer: 
Your log suggests that rasbpian server names (like `mirrordirector.raspbian.org`) are resolved to the IP address `10.0.0.1`, which is [private IP](https://en.wikipedia.org/wiki/Private_network#Private_IPv4_address_spaces). At the very least you don't seem to have a working DNS server, but it's entirely possible you have unrelated network problems which prevent the RPi from reaching it.

Try using google DNS `8.8.8.8` instead of whatever is configured now (put `nameserver 8.8.8.8` in `/etc/resolv.conf`). If that doesn't help, your network connection doesn't work properly and you'll need to fix it.
Once you have followed Dmitry's advice regarding DNS, if you still have an issue...

Check your etc/hosts file to see if you have domains pointed at 10.0.0.1. Unlikely but worth looking.
or
Ping google.com to see if you are getting out there