Why my videos are shown in some browsers but not in others?
Not all browsers support the same video formats, for example Firefox can play OGV and Safari H.264, in all browsers with support for Flash player can be used the FLV format.
The recommendation is to define multiple tags in the video player, for different formats of the same video.
The autoplay behavior was intentionally disabled by Apple:
In Safari on iPhone OS (for all devices, including iPad), where the user may be on a cellular network and be charged per data unit, autobuffering and autoplay are disabled. No data is loaded until the user initiates it. This means the JavaScript play() and load() methods are also inactive until the user initiates playback, unless the play() method is triggered by user action.
My videos in FLV format are supported by all the computer's browsers (including Safari) but not in my iPhone with Safari too.
The FLV is a video format proprietary of Adobe, and supported by applications based in Flash. The Apple gadgets (iPhone, iPod and iPad) are restricted for flash player, so the FLV videos can't be the only one video source, if you want to share your information for iPhone's users.
Why in some versions of Internet Explorer, the interface of video player is a little different?
The main difference in the interface is relative to rounded corners. The rounded corners are achieved with the CSS 3.0 standard but some old versions of browsers have not support of CSS 3.0.
But don't worry about, the main features are all available and the tendency is to reject the obsolete browsers.