Does there exist a public method of completely dumping TWL cartridges, including DSi-specific regions of the cartridge, and what ways exist of running these titles in DSi mode either on real hardware or within emulation?
Decrypt9 (on 3DS) can dump DS/DSi ROMs.
Yes.Can I get a confirmation that Decrypt9 will do complete dumps of DSi ROMs, including DSi-specific memory regions?
There's no indication here that these tools are able to dump TWL software in their entirety, and the information I've been able to gleam from searchng around (e.g. GBATemp's old iEvo review) tells me that most DSi romdumps floating around the internet are missing the TWL-mode regions of data.