One is often led to think that The Bad Batch were created in error, but that view is clearly in error.
In the first episode of The Bad Batch, Tech says their genetics have been manipulated so as to create troops with highly skilled abilities, force, intelligence, leadership, aim, and so on. Omega seems to have their multiple skills.
The Kaminoans were tampering with clone DNA to create something new. Why was that? To what purpose they did it? By whose orders did they do it?
OK, having watched through the series again over the weekend, I do think that we can answer some of your question and at least address the rest.
Yes, Nala Se deliberately developed the Bad Batch, presumably from Omega's genetic material.
It's where I was created. You all were here too. Your mutations were enhanced in this room. Experimental Unit 99 began right here.
-- Omega, Bad Batch S1E15 "Return to Kamino"
It is suggested that Lama Su gave the order (see below). Although I suppose it is possible that Nala Se just went ahead and made these clones (or at least Omega) on her own in secret, and Lama Su merely found out about (and approved) the project at a later date.
Most in Tipoca City don't know about it.
-- Omega, Bad Batch S1E15 "Return to Kamino
We know that the Bad Batch were created in Nala Se's secret lab on Kamino. We also know they were tagged with the "CT99" number. This suggests that everyone else was supposed to believe that they were defective, again presumably for the purposes of secrecy.
This we don't know exactly. We know they wanted enhanced clones as part of some plan to ensure that their position as creators of a galaxy wide army continued, but we don't know the details of this plan. It's likely that at the time, the Bad Batch were created as a prototype for further enhanced clones that could be sold to the Republic.
However, we know that after Order 66 and early in the rise of the Empire, the Kaminoans come to see the enhanced clone project as vital for their own interests.
We must ensure our clones remain essential... If your experiment can yield a superior clone, it will secure our relationship with this Empire.
-- Lama Su, Bad Batch S1E03 "Replacements".
This same conversation includes Nala Se pointing out that the original DNA samples they have from Jango Fett "continue to degrade", meaning that they are likely unsuitable for the Kaminoan's purpose, and that they would need the Bad Batch to continue their work. Lama Su then states that,
We only need one. Our survival depends on it.
Considering the actions taken by Lama Su in trying to recapture Omega, and only Omega, it's clear that Omega is the one that they need.
The Kaminoans don't create without a purpose. You all have one, so what's hers?
-- Cut, Bad Batch S1E02 "Cut and Run"
How producing enhanced clones would save the Kaminoans at this point was not openly addressed, and with Lama Su's implied death during S1E14 "War-Mantel" we are left with only Nala Se who would probably know the answer to that.
From the discussions between Lama Su and Tarkin throughout the series, we could conclude that the Kaminoans felt enhanced clones would be unarguably superior to conscripted/enlisted soldiers, ensuring that Kaminoan clones remained the front-line troops for the Empire. During the creation of Clone Force 99 this is likely to be the Kaminoans' motivation.
However, given that 80% of their enhanced clones had already defected by the time Lama Su claims, "Our survival depends on it," and Tarkin's increasing hostility towards continued cloning, it's obvious that the ability to create new enhanced clone soldiers is of no interest to the Empire.
Lama Su is not an idiot, so this might suggest that the actual reason for wanting to continue the project (and therefore their need for Omega) is something else. It is possible that this other, unknown reason might have been something the Kaminoans were aware of much earlier and was therefore a factor in the creation of the Bad Batch in the first place.
One minor, but pertinent tangent that should be addressed:
I further analyzed Omega's genetic profile and discovered she has pure, first-generation DNA.
...
While our genetic structure was modified for growth acceleration and obedience, Omega is a pure genetic replication. -- Tech, Bad Batch S1E09 "Bounty Lost"
Why does it have to be Omega that they develop new enhanced clones from? Jango Fett was not the only bounty hunter in the universe. If Omega's DNA was completely unaltered and the existing enhanced clones had demonstrated unfavourable traits (such as a high rate of desertion), one wonders if it wouldn't be less hassle to start again with a new donor. Unless the knowledge they had of adjusting human DNA was highly specific to Jango's own genetic code.
Also, if a "pure" sample of Jango Fett's DNA is required, then the Kaminoans could presumably take it from Boba. We know Boba was out of prison and working as a bounty hunter by the end of the Clone Wars and moving in the same circles as Cad Bane. Although Tech suggests Boba's whereabouts are unknown, it's almost certain that Cad Bane could have found him - and they possibly could have paid Boba to be allowed to take a blood/tissue sample for less than they spent trying to reclaim Omega.
This may be an oversight on Lama Su's part, but conjecturably it may indicate that Omega is special in a way Boba is not. We know Omega has demonstrated heightened empathy, awareness, and strategic reasoning skills, these might be signs that her genetic code has been enhanced as a basis for a new generation of better soldiers, but it could also be a side-effect of the fact that she was raised "normally" (i.e. no growth acceleration).
Edit: Season three shows that Omega was deliberately created to test passing midichlorians to clones. The entire purpose of "Project Necromancer" was evidently the resurrection of Palpatine in Rise of Skywalker.
