Mild Spoilers Below?
How Ax's morph works: Excellent question, no idea. Hypothesis: Ax acquires the DNA of multiple people and is able to combine their DNA into one set, rather than four. (I suggest this, rather than that he controls which part of who's DNA he uses each time as 1) his human morph is consistent, and 2) in book 6 he acquires Jake's DNA again ). I'd like to think there is a time limit for the acquiring of each person, but since in TAC it seems Elfangor did it over time, that's probably not true
Somehow he can at least to an extent decide what genes get in this new DNA set, since he "choose" to be male.
But canonical details of how that (and in general morphing) works are absent
Why they don't use it to make new morphs?
Can't be done.
An organisms genetic structure is complex, precise, and it's particular expression vital to the functioning of an organism. If you tried to make a griffon or something you'd have to deal with the expression of a different number of chromosomes from the bird and the lion, lord knows how that would function (Down syndrome griffon???) Beyond that, how is it going to function? And eagles wings should be in the place of a lions forelimbs, but a griffon has them on the back, how does all of that coordinate, how could either's respiratory system support this cross species thing.
You see stories about human tissues being grown on mice, but those aren't functional organs, the mouse can't hear with it because has no neural connections, and can't move it beyond anything embedded in it's back. Tying a bunch of organs together does not an animal make.
Even the centaur thing, which an Estreen can kind of pull off, Cassie, book 1, probably wouldn't work beyond the transitions of morphing. Even being have horse, Cassie probably tripled her body mass, a human heart won't sustain a horses body. And if you but a horse heart up top, it'd probably be too much for the human parts. And if you have both, well I can't see you integrating them.
Ax's technique modifies DNA within species, DNA that is very similar already to make an organism that functions the exact same why as it's precursors, but to make new species by crossing two or more would probably kill the attempter, be it by chromosome incompatibility or morphological non functioning.
Of course, that bears question, can you cross a horse and a donkey (though why would you), and if so, then we start typing into how different animals have to be before this technique is a non-option