signal out of range

Boot the system into Safe mode by holding the shift key down at startup. This should load only the core elements of MacOS, and allow your system to boot to a standard 640x480 or 800x600 resolution with your old display connected.

Use the Finder's 'Find' function (in the file menu) and search for 'windowserver' (no quotes of course). There should be 3 entries - two com.apple.windowserver.plist files and one that also includes a bunch of numbers/letters. Delete all three, then reboot the system normally, again with your old monitor attached. It should then correctly detect the display and give you a range of resolutions appropriate to it.

The TV should work, but I suspect without a dedicated input for computer systems, you'll be stuck with a relatively narrow range of basic resolution settings because the mini won't be able to tell what the display is, and would default to a narrow sub-set of options the TV would otherwise be capable of displaying.

Now you have the system working with the standard display, try rebooting with the TV connected. You should see something like a 640x480 or 800x600 resolution after logging in, though if the system reverts to a distorted image, you may need to refer back to the last message and boot into safe mode and remove the windowserver files as suggested.

Once the system is running and you have a stable display, open the Displays preference pane and look to see what resolutions are listed. If there are a number, be careful to ensure you select only those that fit the correct type (ie, NTSC in North America, PAL for the UK etc). Likely you'll find that much above 800x600 the screen will look pretty fuzzy, particularly on text. This isn't true of all TVs, but generally the ones that work well at higher resolutions are those which are designed for use with computers and which carry dedicated inputs for that purpose.

You can also try experimenting with one or other of the utilities which are intended to allow wider display settings options (SwitchResX http://www.madrau.com/ or DisplayConfigX http://www.3dexpress.de/ ) but neither are all that easy to use.

In the event that you hit a setting that causes the system to fail to output a usable display, rebooting into safe mode and resetting the display resolution there or removing the windowserver files if that doesn't work should get you back to an operable system and allow you to keep experimenting.

The description you give of the display really does sound like a familiar problem when the system is set to a display mode (often incorrect frequency) the display itself can't handle. The windowserver files mentioned before hold those settings, hence deleting them returns the system to default, which should (but doesn't always) work.

If you have removed those files and not had any improvement, the system may be incorrectly identifying the display and setting incorrect display parameters. You could try booting the system with no display attached at all, and waiting sufficient time for the desktop to be opened, then reconnecting the display. This can help resolve some issues of this type.

I'm not sure what you mean exactly!

If you have a login set for your system, you would be best to temporarily disable it. You can do this in the Users preference pane under Login Options (it may request you to authenticate with your admin password to unlock this preference) and then set the system to automatically log in as yourself.

Not necessarily, though I'm not expert at resolving display issues of this sort so it may be that one of the other regulars will have other thoughts for you to consider.

Well I've not come across a display, whether dedicated monitor or TV that did not work even if only at very low resolutions, so there is certainly every reason to hope!

I notice that at the outset you mentioned a splitter cable - do you have anything plugged into the DVI-side of that cable while trying to output to the TV? If so, that could cause problems, and it might prove worthwhile disconnecting it. I also recall discussion here about problematic results with splitter cables before.

If your TV has composite or S-video? If so, you might try a $20 adapter to convert the DVI output on the mini to S-video and composite. I use one of these with my mini and a widescreen TV that has no computer port and while it doesn't render impressive results, at 800x600 it is excellent for Front Row and DVD player.

Indeed it would, but it may be that the cable is the problem. At the very least, it is worth attempting a connection using a simple adapter, if only to eliminate the splitter as the cause.