Re battery pack - the turbo duo/pce duo could be used as a portable machine as you could buy an optional battery pack to use with it. This pack could not be used with the duo-r or rx though.
None of the japanese (ie pce) duo's will run arcade cds without the arcade card duo (not pro..that's for cd rom2 systems)
The exact changes - the duo was black plastic, the duo r and rx white plastic and more sreamlined in design. The duo had the battery pack option the others don't. The duo has a headphone jack on the side of the system, the others don't. The rx differs to the r purely in streamlining (again!) and the six button pad as standard.
You can use the 6 button pad with any pce system. It has a switch on it that allows you to switch between 2 or 6 buttons so you can use the pad for any game on the pce. Be aware though that most 2 button games don't like it if the 6 button option is selected (they'll typically pause unpause v fast...or similar funky madness) but a quick flick to the 2 button option solves this
Re any more stable than the others - the cd half of the duo is known to sometimes stop working due to overheating/overuse...like most cd consoles the more you look after it the longer it will last. I personally think the rx is the one to have...its more sturdy, nice design, good lifespan etc.
Re price/rarity - none are overly rare I suppose, perhaps there are less rx's about than regular duo's...but as with all things pce if you wanna get a nice boxed, wear and tear free duo/duor/duorx you won't find it immediately. Price..duo the cheapest, then duo-r then rx the priciest..but the difference for nice examples of each isn't much maybe the rx is 20-40% more than a duo.
But I guess I play japanese systems most of all, mostly retro too...md, saturn, pce, neo of course. Certainly with regards to the pce/turbografx japanese is the only way to go, seriously dude. There's just so much more good stuff, tons of japanese exclusives, and it all looks really good too if you know what I mean...sleek hardware, gorgeous covers..its a love thing ;D