I'm hoping to use the CDTV as my main 1.3-era box, as it looks so much cooler and works well with the 28" LCD screen that it's connected to 
Have you tried on a 50" LCD TV? It's marvellous. ;-)
Are there also infrared joysticks that work the same as a cabled one?
Never heard of any, but I'm searching one of those, if they exist. :-D
Come to think of it - is there an easy way to make an adapter to plug an A4000 keyboard into a CDTV? I've got one, but the plug is a little different.
CDTV keyboards, have different pin outs. I got mine in good condition from efunzine.com for about 25 euros. It's the original black CDTV keyboard. 8-)