Your problem is faststart problem, it happens in mp4 format. We can consult you best formats for streaming over internet, also we can help you solve faststart problem.
We have worked with FFmpeg for 5 years. My transcoder is based FFmpeg with some customize to more control and stable.
We have a lot of experience with live streaming. Especially WOWZA, Darwin, Red5, Live555... We have build a secure module(authentication module) for secure streaming.
We have also sucessful developed a Video Platform, VOD(video on demand) and Live Video. We have a lot of experience in live transcoding and streaming video. Your solution supports many inputs type as RTMP, UDP, RTSP, RTP, HTTP...output is encoded in H264/AAC over RTMP, UDP, HTTP...target devices are PC, tablet, smartphone( iOS, Android, Windows phone), smarttv.
We can streaming from Camera, Camrecoder, DVB-S, DVB-T video capture card... to web, smartphone, IPTV. We can consult you build a good video streaming system.
Let me help you.