I had precisely this problem with the first GH2 I purchased. Everything worked fine with the Sony Nex and the Canon 5D, but the GH2 would only permit playback. I sent it back quickly for a refund. I eventually bought another one (by which time the price had increased £150) and exactly the same thing happened. (insert all sorts of expletives HERE).
It turned out that the problem was caused by the crappy bloody HDMI converter. Once I had swapped that stupid huge great gangling piece of crap board that most people seem to buy (the one that arrives in the 1/2 kg metal box) for the tiny, lightweight plastic-cased MiniHDMI, the problems were solved and the GH2 was able to deliver live video to the downlink.
THIS is the Mini HDMI converter I bought >It has since shown up everywhere, including Amazon I believe. You will find more on this subject on the
Sony Nex-5n - downlink thread and there is a tutorial on wiring and stuff
HERE.
You have slightly misunderstood the frame rate thing. If the GH2 is set to record 720p25/50 it will automatically change its video record frame rate/resolution setting to 1080i50 as soon as you connect an HDMI cable - even if there is nothing on the other end of the cable. If the camera is set to 24fps then it will remain at 24fps when using the HDMI socket. Video is output in all cases, regardless of the frame rate/resolution setting. After all the hassle of getting a downlink signal and losing £150 on the price of the GH2, all because of that crappy bloody HDMI converter, I ended up using a separate monitor camera next to the GH2 anyway because I wanted to record at 720p50.
The Zenmuse turns all that upside down with its various restrictions but at least the HDMI converter built into that also gives you a pretty neat remote Shutter Release/Record as compensation.