I would like to play a video from inside the rendered page with an iPhone/Safari.
When I put a <video>, it's always opened and played in a separate full-screen Quicktime window. Is there a way to play the video directly from the page?
(a similar question http://stackoverflow.com/questions/751627/iphone-play-video-not-in-full-screen-mode suggests it's not possible)
No, with the current SDK compatible media will always play in full screen and there is no way to avoid that.
The Safari browser on the iPhone/iPod touch are unable to play video or audio within the browser because of the restraints that Apple has placed on it. They do not allow streaming media or Flash compatibility because it would defeat the purpose of creating apps (people would just rely on flash based apps and other games).
It is not possible to play in-browser video or audio without a plug-in such as seen in desktop browsers - that being said, you can have your video be played on a button press within the browser and have it go to the same page on finishing.
Yes, it's possible to play video within the browser, but as you said, in the Quicktime fullscreen application, and the HTML5 sound API is not available (in iPhone OS 3.x).
But in the coming iPhone OS 4, it will be possible (used for iAd), so you will have to wait until this summer (or play with the beta SDK now).