Auto Start Popup Player

When I click on a presentation, another page comes up on which I need to click on "Popup Player" in order to get it to play.

Is there a way to get the presentation to play immediately when one clicks on the name of the presentation?

www.illa.us/listen/sermons?catid=86

Also, is there any discussion on the various features and pros/cons of using the various players like JW Player, Real player, etc.?

Have a look at the "Columns" tab in the SermonSpeaker options. You can set a lot of information there. One of it is the player which you can show on the list and the single sermon page.
In the "Global" tab you can also set what should happen if someone clicks the sermon title in the sermon list. One option is that it controls the player.

As for the player: Assuming you recorded mp3 files, I would use JW Player, Flowplayer or Pixelout. Which one is probably more a question of design. JW and Flow have more functionality, however if you use the icons layout, it will only load the Pixelout player because the others just don't look nice there.
Real is only interesting for those using the special file formats supported by this player.