Re the Empire building multiple Death Stars.
I read a ton of Cold War history and basically it’s the learning curve that is the bear.
Basically the US and the USSR poured tons of money into programs just to get a rocket to go up, off of the launchpad, without it exploding. Post World War 2 to Sputnik going up is 12 years of R&D on both sides of the globe. That’s to get one satellite up in orbit. From 1957 - 1969 we put people up there, did space walks, circumnavigated the moon and landed on the moon & came back.
There’s time and expense after the learning curve was solved but the real bear for both countries was the foundations of rocketry.
At the end of Revenge of the Sith you have a proto-Death Star being built. We have no definition that this is the one that we see 20 years later in A New Hope. We just see one being built. If you adhere to the learning curve thing this one could have easily blown up at some point while being tested or even sabotaged (I always like the line in A New Hope, “there’ll be no one to stop us this time” could refer to sabotage and doesn’t have to be rebels doing the sabotage, there were real division and competitions in the US between the Army and the Air Force over who gets what missile, we see that kind of thinking in Krennic v Tarkin in Rogue 1).

On to DSI, DSII and starkiller.
Since we made nuclear weapons all we have done is made bigger, badder, stronger iterations of the same.
There’s no reason that the Empire would not do the same thing. A ton of R&D dollars went into the first one and it basically did what it’s designed to do (zap a planet into oblivion) so there’s no reason to think that they wouldn’t try to refine it.
Just read “Red Moon Rising” a quick read on what was going on with rocket programs in the states and USSR when Sputnik went up, it’s a good place to start for anyone interested in this sort of thing.