Let's say Alexander the Great lives longer and finally consolidates his new empire instead of conquering more, can he achieve the level of power and prestige that the Roman Empire had in OTL, under the right conditions?
Achaemenid Persia was able to last two centuries.The biggest factor is the Med. For the Romans it was the unifying highway, and they only really expanded much beyond it as they became more of an imperial machine. This is why the empire didn't grow in stages of removal from Rome itself, but rather in stages of removal from the Mediterranean. And even so, their 'hot spots' we're almost always the parts of the Empire farthest from the Med, ie Rhine/Danube, Mesopotamia and Britain.
Alexander's empire would be helped by the Med on it's Western half, but would be forced to rely on overland/river systems from it's center eastwards, including it's capital and principal satrapies (assuming stays Babylon).
"... and his children are competent enough"
Slight problem; I know that era's Greeks had, um, curious sexual mores, roughly translated 'boys for fun & girls for heirs' but, IIRC, Alexander shunned the latter...
Okay, we may assume he might marry a Princess or such to consolidate a take-over, but still there's a fair chance he'd die heir-less, or the kid(s) would be killed in a palace coup...
I don't think Alexander had to "institute fundamental political reforms" concerning the succession rules.(That said, it was very astutely noted on this board - years ago - that when he was the age at which Alexander died, Augustus had not yet consolidated anything much. Even after gaining power, his initial political strategy was "become consul again and again and again". Only when he fell severely ill, and could only barely prevent rivals from exploiting his weakness and ousting him, did he finally decide to institute fundamental political reforms. Which gave us the Principate. So, given those facts, I refuse to believe that Alexander could not have followed a similar path. It may not be the most likely outcome, but it's within the bounds of realism.)