Welcome to the forums. Let me try to help with your questions. Short Answer: Yes, but it's not as easy as developing a Flash/AIR application.
"Is it possible to create a video chat application for IOS (Native Objective-C) using Wowza?"
Yes. It is possible. But, you would have to write an H.264 encoder from scratch. ;) Otherwise you'll have to use some additional library. There has been work done porting existing implementations such as FFmpeg, or Live555 to iOS. There are RTMP streaming examples such as "Live RTMP Streamer" (from FOX-IT) available in the app store for over a year, the Livu API which is available for licensing. There are many other commercial iOS Video Chat solutions. Some of these may use Wowza, but I don't know of any specifics.
Other ongoing projects include OpenTok, WebRTC, Dropcam, aftek-iphone-RTMP-library, ffmpeg4iphone.
"How to send IOS camera and Microphone video/audio live stream to Wowza server?"
Simply generate a stream in one of the Wowza supported codecs and protocols.
"Is it possible to do chat with Wowza and IOS without using any RTMP library for IOS?"
You could use RTSP, RTP, or perhaps something entirely custom. But, I am not aware of any native support for protocols other than Apple HTTP. Additionally it is my current understanding that the h.264 hardware encoder on iDevices currently available is not designed for real-time as it must finalize the encoding before it can be decoded. If that is true, this limits you to software encoding, or Apple HTTP.
It is my understanding that the Apple Requirements do not allow RTMP to send H.264 data that is sufficient for a Video Chat application. (
source) Perhaps other codecs such as H.263 or VP6 are possible using RTMP, or H.264 with a different protocol.