I figured it out just now. I tried a different browser, and that worked. Then I realized it’s the adblock extension in Chrome that caused it. Removed the social part of the adblock extension and it now it functions properly. I feel pretty dumb right now.
It was happening while being the only plugin installed on a fresh install of WP. I did what you said anyway, no difference. Same issues. Can’t move the buttons around, and it’s missing the majority of the buttons as well.
After activating and connecting the WP blog to the Jetpack plugin, I can manage things from the wordpress.com website. There is nothing available, as you can see from the screenshot below.
I completely removed my hosting, and setup fresh and new. STILL have this issue. Installed WP 4.2.4 with the latest Jetpack plugin update on PHP 5.6 FastCGI.