Q: Connect my 21.5" Late 2009 iMac to HDTV?

Use a mini-displayport to hdmi adapter adapter and a hdmi cable or a mini-diplayport to hdmi cable to connect the imac to the TV.

Set the TV to the input you connected the HDMI cable to.

On your computer, if you open the Displays preferences and do not see an Arrangement tab (or there is an Arrangement tab that shows only a single monitor) then there is something physically wrong with your connection.  Try clicking Detect Displays just to see if the computer can now sense the presence of the connection. Check your cables and make sure the mini-displayport plug is pushed firmly in so that essentially all the metal part disappears inside the socket.  Mini-displayport plugs can be a little tricky in that you need to make sure the plug is fully seated in their socket.

If you do have an Arrangement tab, and it shows two screens, then click Gather Windows to bring the TV's Displays preferences on to your main screen if the TV is not yet showing an image.  Set the proper resolution (probably 720p, 1080i, or 1080p).  Set the refresh rate (probably 60Hz) if that preference is shown.  You should see a picture at this point.  I  recommend turning off mirrored displays if it is enabled (more about this below).

You may have to play with the underscan/overscan settings on your TV and the displays prefs for the TV if the picture is too small or larger for the TV screen.

Also, you need to set your Sound preferences to output to the HDMI connection.

And if running Lion, do not run in Lion full screen mode.

Arrangements, Menu Bar, Extended Displays, and Mirrored Displays

When you have multiple monitors (and the TV is considered just a monitor) only one monitor is designated as the "main" monitor.  That's the one with the menu bar.

If you look at the Arrangement tab in your Displays preferences you will see both monitors.  One will have a little white "menu bar" on it.  That's your main monitor.  You can drag the little menu bar to the other monitor if you wish to make it the main monitor.

You cannot make both monitors main monitors.  However there is a third party app called MenuEverywhere that allows you to do that (it has mixed reviews).

Your two monitors can be spatially arranged with the Arrangement in Displays as well.  Drag the monitors around each other to define their relationship.  Then you can drag the mouse or windows across with in that "space".  This is extended desktop mode and I recommend that to be able to use all the available screen real estate.  However there is a checkbox in the Arrangement to run in mirrored mode.  Now the two images are identical on both screens albeit the max screen size is determined by the smallest screen in the mirrored set.  You would of course have the menu bar on both screens since they are mirrors of each other.  But I think it's a waste of good screen real estate.

Playing videos in Extended Display Mode

Flash videos are normally played in your browser using the adobe flash plug-in.  If you click the video to display full screen the video will expand on the monitor containing your browser window.  Therefore you need to move your browser window on to the desired  monitor to view it full screen.

Many stand-alone video players, for example, QuickTime, operate the same way.  You need to move the video window on to the desired monitor.  There are exceptions however.  The more powerful vlc player has a a preference to allow you to specify which monitor is to be used for full screen display.

I think I did, 1 at least.

1) Any current hdmi cable will do like the ones at monorprice or what you could pick up at your local drugstore.

2) You use vlc, so do I.  I play the videos on my TV but I keep the vlc controller on my main monitor.  So I don't sht off the main monitor.  I do usually dim my main monitor with isolator (a menu item I that I click to toggle the screen dimming).

I have no idea about the apple remote to adjust volume.  I just use my TV control.  I also don't know what you mean about using the remote to "scan".

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I guess for full disclosure my setup is a mac pro, not an imac, but that should not make any difference for this kind of an application.

1). See the following:

About Mini DisplayPort to HDMI adapters

So yes, it passes audio to the mini-displayport.  The adapter I linked to explicitly says it passes audio.  Don't get any cable that doesn't explicitly say it doesn't have audio pass through.  They all pass the proper video.

2). Why this obsession of blacking out the imac monitor?  No, you cannot do it.  The imac has to stay on obviously to feed the TV.  The best you can do is dim it as far as you can dim it.  I don't understand why this is not a "viable solution".

After a little more thought there may be one way to black out the imac screen.  That is to disable it and make the TV the main (and only) monitor.  I think you can do this with SwitchResX.  I know it can enable/disable monitors and change which monitor is the main monitor (the one with the menu bar) so I assume you could also do it to the imac monitor as well.  So that would effectively remove the imac as a monitor and it would go dark.  Later you use SwitchResX to re-enable it when you want to use it again.  These configurations can be defined as a menu (menu bar and/or contextual) and/or a hot key.  So once configured in SwitchResX, switching monitors is a simple as pushing a hot key or selecting a menu item.

Sorry, apple is wrong.  There are no Displays system preferences to fully disable a monitor.  That's one of the reasons SwitchResX was created.

You could set a monitor to an invalid resolution and it will go black but not as far as the system is concerned.  So based on the Displays Arrangement you can move your mouse off the screen you can see to the one you can't and that gets confusing with you wondering where the mouse curser want.

SwitchResX can truly disable a monitor just as if you pulled its plug.  The mouse cannot get "lost" on the disabled monitor.

This is why we reference the so called apple store "geniuses" in quotes in these forums!

See my initial (long) post on what defines the "main monitor", i.e., it's the one with the menu bar.  And you can drag that little white menu bar from one monitor image to another in the Displays Arrangement window to define what is the main monitor. 

No checkboxes except to specify mirrored mode and one to put a displays menu  in the menu bar to do some of the same things the Displays preferences does (and also to launch the Displays preferences from that menu too).