Out of curiosity do you have issue when running Iceweasel? I had some issue with my IDE (PyCharm) and a few random web applications that just hated Iceweasel. I just self installed Firefox.
I'm not the parent but I've never had issues with Iceweasel. There was a time when the user agent string did not contain "Firefox", but it does now [1] (that might have caused you problems?).
And if you're using wheezy, there are backports to the latest Firefox (iceweasel) packages. I usually see the update on the same day or the day after the official Mozilla release.
[1] since 2009 http://bugs.debian.org/cgi-bin/bugreport.cgi?bug=399633