@Dualizing for profit: Generally speaking, you're better off just getting to a high value quality (e.g., most of the 8-letter ones). So your Dreamer's Flange example should really just get to Perilous or something and then sell it. Not much reason to go through a bunch of dualizes to raise the resell Inheritance. I mean, yes, you'll make more money but it's not really worth the time spent.
The upgraded Rare weapons (like Dreamer's Flange) all have such a high base resell value that you really don't need to do much to make them worth something. If you're just trying to make a profit off of Rare equipment, you might want to dualize a few times and try to end on a high Resell Multiplier Quality.
To make the most money, you only care about Resell Inheritance for the in-between steps and you want the highest Resell Multiplier for when you're ready to sell the item. For example, if you're going to Dreamer's Flange +4 then steps +1, +2, and +3 you only care about the Inheritance, not the Multiplier; that final dualize to get it to +4 is when you want a good resell multiplier Quality.
I guess I can make a list of shards to use but it's not something I'd really try to adhere to strictly since you may not have the right qualities for it. Just know the Inheritances and Multipliers of each option (the game tells you this so you can always just check it) and figure it out from there. I generally just took most equipment down to a 5-letter Quality (Rigid, Noble) and then up to 7-letter (Magical) or 8-letter (Perilous, Poignant, Dazzling, Fabulous) depending on what I had.
And do remember to use Miso Stew as your food during this. It makes the entire process pretty fast as it takes 2-4 fights for most the tempering.
The upgraded Rare weapons (like Dreamer's Flange) all have such a high base resell value that you really don't need to do much to make them worth something. If you're just trying to make a profit off of Rare equipment, you might want to dualize a few times and try to end on a high Resell Multiplier Quality.
To make the most money, you only care about Resell Inheritance for the in-between steps and you want the highest Resell Multiplier for when you're ready to sell the item. For example, if you're going to Dreamer's Flange +4 then steps +1, +2, and +3 you only care about the Inheritance, not the Multiplier; that final dualize to get it to +4 is when you want a good resell multiplier Quality.
I guess I can make a list of shards to use but it's not something I'd really try to adhere to strictly since you may not have the right qualities for it. Just know the Inheritances and Multipliers of each option (the game tells you this so you can always just check it) and figure it out from there. I generally just took most equipment down to a 5-letter Quality (Rigid, Noble) and then up to 7-letter (Magical) or 8-letter (Perilous, Poignant, Dazzling, Fabulous) depending on what I had.
And do remember to use Miso Stew as your food during this. It makes the entire process pretty fast as it takes 2-4 fights for most the tempering.