If Charles Martel decide that only the senior son/ his most capable son would hold (most of the) political power, that he decided against dividing his realm?
History wasn't a Crusader Kings game, when people could "choose" to change the laws, if it humoured them.
The concept of primogeniture was unknown when it came to kingship, critically when the Frankish political succession was about dividing the fisc and sharing kingship or at least power.
Would have he done that, Carloman would have certainly rebelled, supported by many Frankish (and possibly ennemy of Charles as well). Of course, it could have ended with Pepin III victory, but it would have weakened enough his dominion when Peppinid lacked legitimacy enough that they had to crown a Merovingian when they came to power.
As for Rome, Carolingia fall under a whole net of factors.
Abassid economical collapse, that not only harmed its own but provoked a push for Scandinavian raids; change of social paradigm towards feudalism (Basically, Carolingian tried to introduce a generalized vassality, instead of a limited one, and it backfired); no more "good" conquest to do forcing to a re-focus on inner matters; climatic changes and lower agricultural productivity, etc.
The look of the empire was good, but it's like gold on rotten wood. Sooner or later, these issues were to appear.
ASAIK it is the first 'peak' of Europe since the Roman days.
Carolingians were really about gloryfing themselves (admittedly they had to, because their legitimacy was somewhat dubious at first), and that they did a good job getting rid of Merovingian archives.
Truth is that without Merovingian Francia grounds, Peppinids/Carolingians didn't had a real chance to pull an empire.
If the empire can endure or sustain in a reasonable way. Can the renaissance come back earlier that the Empire pick up the lost science/ technology of the antiquity?
Carolingian Renaissance was essentially an elite renaissance, with little to no impact on imperial structures. It wasn't even tought in a different way : it was about elites being structured imperially.
Technological or Science wasn't their main focuses (not that it was "lost". Without countring Byzantines or Arabs there; it was present just roughly unused), and these didn't had a real gap between Late Roman Empire and, say, XIth century.
People didn't just became stupid in 476.
I do not know too much about history in those days, I wish it is not too ASB.
Well, it is, but given the ammount of BS you can have on this period, I don't think anyone would blame you.