Flight numbers are the same for each plastic variation, but in my personal experience K3s fly more like 1/0/1/1 once they've been loved over a few rounds of play and tree kicks. Currently bagging a K3 Hard Berg, but haven't beat it in enough yet to tell if the same is true for the new plastic blend. I've never thrown one in K1 Soft, but I bag a K2 which should theoretically be the same expected flight; less stable than K1, but more stable than a beat K3. Overall I find their stability to be, in order of least to most stable: K3, K2/K1 Soft, K1/K1 Regrind/K1 Glow. I know some people have K1 Glows that fly more OS, but I never noticed much difference between those 3 K1 variants; for me they all fly as expected and the 2 fade is reliable.
In regards to dome, I've definitely noticed K2/K1 Soft is more prone to doming up in the middle, but I've seen K1s that do the same just not as regularly. Most K3s I've seen are flat to puddle topped. I love my puddle topped K2 Berg, but don't like throwing K2s with dome; they feel much deeper in my grip compared to puddle tops.
Lastly, K3 Hard feels exactly the same as K3 except it's firmer. I can bend a K3 Berg in my hand and it will have some flex, but K3 Hard is very stiff with little flex. Same feel as K3; overall I've found K3/Hard Bergs feel "smaller" than others; I've noticed the same thing about Rekos, but YMMV. If you're wanting a Berg without that dome in the middle then I'd suggest keeping an eye out on OTB for the next restock; they do a great job of describing the dome/firmness of every disc and it's great knowing what you're gonna get. My first K1 Reko had dome and I really disliked the disc until I got a puddle top K1 Reko that quickly became my favorite disc.