Firefox 34 is sufficiently different as to require a separate DLL to support it, much like past Firefox versions. This is not surprising.
If you'll check the support docs, you'll notice that TestComplete 10.50 does not *officially* support FF34. The support is currently unofficial but (I've learned) there will be a patch coming out in a week or two to provide that official support.
Please note that we (SmartBear) will support the various combinations of patches to our versions of TestComplete as are listed in the Downloads section for the various browsers.
Eg:
http://support.smartbear.com/downloads/testcomplete/firefox-patches/Note also that if you try to use unsupported combinations, and then ask for help, we shall have to give you a stern look and scold you gently but firmly.
With that said, sometimes you just simply need something to get you going. TestComplete 10.40 Object Spy doesn't recognize FF34. TestComplete 10.50 Object Spy does. I've noticed that if you copy the "unofficial" DLL for FF34 as provided in TestComplete 10.50 (bin/extensions/tcFF34Hook.dll) into that directory for TestComplete 10.40, then TC10.40 Object Spy will recognize FF34.
Will this work for TC10.20? Will this let you run tests against FF34 using TC10.40? No idea, but you can always play around with it.