VisionMobile has done extensive research (97 pages) on this in their Cross-Platform Tools 2012 report. It's available for free after registration. They've found over a 100 different tools and written detailed profiles of the 15 major players. Those are:
Adobe PhoneGap (Apache Cordova) - JS apps with HTML/CSS/JS UI inside a webview
Adobe Air / Flex
Appcelerator Titanium - write JS instead of instead of Obj-C or Java, Titanium did the webview thing similar to Phonegap in older versions
IBM Worklight - basically a commercial enterprise version of Phonegap
Ideaworks 3D Marmalade
Rhodes and Motorola Solutions RhoMobile - Ruby with webview
Sencha Touch - JS/CSS UI framework
Unity - game engine, probably not what you're looking for
Xamarin MonoTouch and Mono for Android - write C# instead of Obj-C or Java
And depending on the used tool, it may or may not have the ability to write platform native code and/or UI if the framework doesn't offer the needed capabilities.
Now coming to your question about how Adobe AIR performs, I would say -
As a user I can say that whenever I see an application is using Adobe Air, I know I'm gonna have problems.
In desktop applications where text-boxes and right-to-left text leaves a huge mess - I usually copy the existing text to Notepad, edit it, and paste it back into the Air-based textbox. The developer's response? "It's the Air platform, there's nothing we can do about it".
In Android I've had even worse problems. One application I've installed recently, a remote file browser, simply replaced all non-Latin characters with spaces.
Some of these are developer bugs, of course, but many are the fault of the platform. Windows has been the first major OS to fully support Unicode and RTL text in a consistent and useful manner. Android has come a long way since 2.1 and its bugs, and new Android versions get it 90% right. But then comes a long an application platform like Air that starts from scratch, reinventing the wheel, and pushes us 5 years back in that respect.