As the surprises tripped one into another, providing rich fodder for the 15,000 journalists who had flooded into Cleveland, it seemed that Mr. Trump had approached the convention in the same seat-of-his-pants manner that served him so well in the Republican primary elections this spring, when he steadily dispatched a field of 16 opponents.

While political pundits spent much of the week tsk-tsk’ing the Trump team for lacking control of the conclave, they have been proven wrong over and over in this unconventional year, in which the former reality television star not only threw out the old playbook but also seems to be operating without a playbook at all.

Mr. Trump’s thunderous and fear-laden acceptance speech Thursday night got a roaring welcome in the convention hall, but the chaos of the past week renewed broader questions about the haphazard and deeply self-involved style that assured his stunning success up to now. Among some Republicans, it has reinforced doubts about whether Mr. Trump, who has built his campaign around whim and impulse, can now harness that into a viable force to beat the Democratic contender, Hillary Clinton, in November — and to run the most powerful country on earth.

“You look at that, and the conclusion you come to is that this is not a serious endeavor to elect someone president,” Stuart Stevens, who was the chief strategist to the 2012 Republican nominee, Mitt Romney, said of the convention. “It’s just about ego.”

Every twist of Cleveland crazy was fed into the instant grinder of social media and news websites, obscuring the messages from the podium to the party faithful: testimonials from Mr. Trump’s wife and children to soften his often belligerent persona; rallying calls to unify his bitterly divided party; and the elevation of his gentle-mannered running mate, Gov. Mike Pence of Indiana.