First, I just tested the download link and it is now back online, you can download the OSX installer from http://downloadmirror.intel.com/24910/eng/PhoneFlashToolLite_188.8.131.52_mac64.pkg. If you are having issues using the Flash Tool Lite, then you could try using the old way of flashing.
Download the latest Yocto image from IoT - Intel® Edison Board Download | Intel® Developer Zone, and then unzip it. Now connect your board to your computer with the J16 micro USB port (and a power supply if possible).
Erase any file that's on your Edison's memory, copy and paste the files of the Yocto image into your Edison's drive.
Now connect the J3 micro USB port and access the Edison's internal Linux as described in IoT - Step 3: Set up a serial terminal | Intel® Developer Zone.
Once you have entered your Edison's Linux, enter root as username and then type the command “reboot ota”. The Edison should reboot and install the image you stored in its memory.
Regarding the email issue, I will make sure to inform the appropriate team so this issue can be resolved as soon as possible.
Let us know what happens with your Edison and if you were able to flash it as expected.
I tried again after I got home and had two usb cables, and apparently that wasn't the issue because a "regular" install still failed, but the manual install did work, so my Edison is up and running now.
That said, the FlashToolLite.dmg link on IoT - Installing Flash Tool Lite on Mac* OS X* | Intel® Developer Zone still appears to be broken to me - it points to an older version, 184.108.40.206 which appears to no longer be available on your server.
You are right the link is down; I was looking at the link in Intel® Edison Board Download which worked for me. I will make sure to also let the appropriate team know about this. Thanks a lot for bringing it up. By the way, if you still have issues with your email, please submit an email to this form. We will try to reply to you as soon as possible.