Any luck with this? I'm having the same issue, also using 4.7. Here is the log info:
8/7/13 9:59:29.515 AM Adobe_Flash_Builder_4.7_Plug-in_Utility_: Could not find image named 'lock'.
8/7/13 9:59:33.006 AM coreservicesd: Application App:"Adobe Flash Builder 4.7 Plug-in Utility" [ 0x0/0x1b65b64] @ 0x0x7ff5e2b3cac0 tried to be brought forward, but isn't in fPermittedFrontASNs ( ( ASN:0x0-0x1b66b65:) ), so denying.
8/7/13 9:59:33.006 AM WindowServer: [cps/setfront] Failed setting the front application to Adobe Flash Builder 4.7 Plug-in Utility, psn 0x0-0x1b65b64, securitySessionID=0x186a4, err=-13066
8/7/13 9:59:33.013 AM authexec: executing /Applications/Adobe Flash Builder 4.7/utilities/Adobe Flash Builder 4.7 Plug-in Utility.app/Contents/Resources/Adobe Flash Builder 4.7 Plug-in Utility.app/Contents/MacOS/Adobe Flash Builder 4.7 Plug-in Utility
8/7/13 9:59:33.014 AM Adobe_Flash_Builder_4.7_Plug-in_Utility_: osError = 0
Kind of. I never actually solved the bug, but I did make it past the install.
Upon running Adobe Flash Builder 4.7 Plug-in Installer it appeared to crash and only load a blank screen with a single unclickable button. It wasn't actually "hanging", and would react when I clicked on the program menu and moused over "Services". This would allow my clicks and keystrokes to register with the program, but only after I mouse over the "Services" menu item. For example, you need to click "okay", then mouse over "Services". When you want to enter text you need to click in the text box, and mouse over "Services". Then you need to type the text you want, and mouse over "Services" to see the text appear. It's a PITA, but you can get through the install by doing this.
After installing the FB plug-in starting eclipse with editors open appeared to cause errors. If this happens, this is because it seems that FB attempts to load when eclipse starts up. Close all editors and restart eclipse. The Welcome page for FB may pop open up. At the bottom there is a check box that will keep that form loading - check it. If the page does not open when eclipse starts, then open up an MXML file. The Welcome page should load, and you can check the box.
I hope this helps you move forward. Also, if you find an actual solution please let me know.