Thanks so much for the detailed list of things to try. I did everything off the list except the following:
I didn't really understand how to do 3c and I managed the first part of 3d (enabled HTTPS) but couldn't figure out the second bit:
Add both *.wordpress.com and *.wp.com to your list of trusted sites under the Security tab. Make sure to uncheck the box for “Require server verification (https:) for all sites in this zone.” http://en.support.wordpress.com/https/#frequently-asked-questions
Everything is up to date. The button links still don't work.
So the answers to question 6:
a) MSi laptop with windows 8, New iPad Air (all updated).
b) On the laptop I'm running firefox 29, on the iPad I've got Mobile Safari 7
c) There are no error reports. On the laptop, if I click on one of the social media buttons or the meetup button it seems like the destination will load in the same window because it goes blank white and says 'connecting' briefly in the address bar, but then nothing happens and you're left with a white screen. If you click back, you don't go back to my site either.
On the iPad, you press hold one of the buttons and get a menu, if you touch 'open' nothing at all happens. If you select the next option 'open in new tab' it works fine.
Interestingly if I go to the full site on my Samsung Galaxy s2 and press hold on the logos, I get a menu where the first option is also 'open' and in this case they do! On the phone I've got the Android 4 browser.