NOAC handheld ROMs

Regarding the Atari Flashback, the Intellivision x-in-1, and others that use a NOAC to reduce manufacturing cost, have the Atari 2600 and Intellivision game remakes ROMs been dumped? And if those ROMs were available, is that something that could be easily emulated (perhaps with a new mapper) or are there other custom chips in those handhelds that would required a lot of research and analysis before that could ever happen?

No, the consoles have not been dumped yet; neither have the individual games in their original ported versions with Atari-2600-/Intellivision-grade graphics. The individual games however have been upgraded to normal NES grade and also to enhanced VT03 grade by the developer, Nice Code Software, and included in various dumped and available plug-and-play consoles under different names. Look for "Project Plug-and-Play" by going to the forums on Bootleg Games Central.

Once the full consoles you have mentioned are dumped, they will likely use mapper 256 or a variation thereof. You already have the NintendulatorNRS source code that emulates it, along with the VT03 enhanced graphics.