How then is this such an easy thing to do with other A/V equipment like my TiVo's (HD & Series 2), as well as with my Sony 400-disc DVD Player? Why would the mini have an issue here if these other devices don't?
Because connecting computers to displays has its own legacy and industry standards, they connect over VGA and DVI, and a computer is a much more universal and complicated device than a "mere" consumer electronic device--sometimes this sophistication works in favor of a computer, sometimes it doesn't, and 1) connecting to "televisions" and 2) getting a properly sized desktop resolution are two of those things that often isn't anywhere near as easy as it should be. When computers hook up to a display there is two way communication, polling and data exchange going on--rather than the dumb, plug in those RCA or components cables of that dvd changer into anything and it'll play. It's only recently that we've even thought of a computer serving in that CE world of audio/video--and sure, you're gonna say, well my Tivo is a computer, right? And it is, but it's a computer that's more narrowly focused, stripped down and designed to make connections over certain broad and long-time standards--usually analog via RCA or component--whereas computers, and especially Macs, have their own history of connecting over analog--it's just that "their" analog is a different standard, VGA. And that brings up the different colorspace issue, i.e. having to transcode from the colorspace of computers (RGB) to the colorspace that most CE devices and TVs expect (YPbPr.)
The "Tivo" equivalent in this case might be the Apple TV--it's designed to connect to widescreen TVs, it has component built-in--and many use it to connect to their older HDTVs (without an HDMI, DVI or VGA input) instead of a Mac just for that reason.
Add in a 2nd layer of complication: that Macs by design are inherently more disadvantaged than Windows PCs when it comes to graphics cards options and tweaking drivers to accommodate component--we don't have the choice, access or options. In your case with the mini, if you want to run OS X, you're limited by what Apple allows the GMA950 to do.
Then add in a 3rd layer of complication: Manufacturers of HDTVs share blame for this--their ability to recognize, properly connect to and share EDID with computers leaves a lot to be desired. It's a two way street--and manufacturers of high def TVs have also been relatively slow to evolve.
Do you think a VGA to component transcoder might work, or is this something that you're suggesting might work only if I plan on connecting my mini directly to my TV's VGA port?
I think you should try going straight VGA to VGA first, see if you can get your desktop to appear at the native res of that plasma, which is 1366x768. That'll tell you if your new mini can function as an extender, directly connected to your HDTV, which is how almost all of us do things. Even this isn't guaranteed, given the fickle nature and limitations of some older HDTVs, and you may be in for a lot of pain (via SwitchResX or DisplayConfigX) merely trying to make this connection.
Based on what happens--then, yes, you might consider a VGA to component transcoder--and some people have had success going through an AVR with one. You'll have to tap into whatever knowledge base there is out there of fellow Denon users because going through an AVR adds yet another layer of complication--that's your fourth--you might not be able to get "1366x768" through, and once it is through, when you switch away to a different video source and then back, your Mac might not be able to "see" and regain sync with your HDTV at that previous timing or resolution again. Some folks who have to deal with this--going through an HDMI switching AVR--eventually find success with something called the Gefen DVI Detective--it holds that certain resolution and timing in memory so the Mac thinks it's still connected.
But--at some point you have to ask yourself is spending more money to try to force these solutions into prescribed notions really worth it long term? And is the hassle configuring it worth it when there are much easier ways, albeit with different equipment and approach.
I think you're in a tough spot: you're trying to drop a new concept--home theater PC--into older, limited equipment you currently have that wasn't designed to "just work together" the way you're currently envisioning. And not only that, if I read your other post correctly, you're trying to integrate it into a "whole house delivery system" that you've already set up and that relies mostly on an older, more out of date, analog "consumer electronics" way of thinking. Those Denon AVRs are expensive, you have A LOT of money tied up in them, and they're wonderful for sound and switching audio--but they're not digital video switchers. They're just old enough (4 years or so) just to have been behind the oncoming digital and high def wave--and AVR manufacturers are traditionally slow to evolve anyway. You're paying that price now as technology, especially video technology, swiftly moved forward.
my speakers are built into the ceiling on my living room and I believe the only way to get sound into them - or out of them, depending on your perspective - would be to run the audio through the receiver.
right, I'm probably not telling you anything you don't already know, but most of us run optical audio from a Mac to an AVR--it's just that our HDTV, AVR and Mac are in relatively close proximity to each other. I think the more common "whole house" strategy around here is to prioritize Macs first, to have multiple Macs and aTVs on a gigabit network, with some carefully chosen wireless devices, everything networked and sharing content. If you want to set up more than one main listening/viewing area--have an HDTV, AVR and speakers set up in each area, with a Mac or aTV directly connected to each HDTV and optical audio going to each AVR. An Airport Express and self-powered speakers can get music to secondary locations, a laptop or an aTV in a bedroom can pull video when necessary.
This strategy allows you to offload as much to the Mac HTPC as you want AND still keep the more traditional CE devices connected to your AVR and HDTV--parallel delivery paths if you will, rather than integrated. One foot in the old, one foot in the new.