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.
Hope that helps