Does Minato Namikaze want to do everything on his own? I feel like Minato died because he tried to solo the Nine Tails, which worked, but at the cost of his life. 

Hiruzen, the leaf shinobi, as well as the Anbu Black Ops, were all fighting the Nine Tails (and succeeding) until Minato came and teleported the Nine Tails like 10 miles away (in fact, they specifically said ”the Nine Tails is out of the village”) and tried fighting it by himself right after he fought Obito. 

To me, this sounds like a "try hard" shinobi. Did Minato want to do everything on his own?
Minato did not want to do everything on his own - he was forced to do everything on his own. During the Konoha attack, Minato was performing his duties as:

A husband/father, who was protecting his family
The Hokage, who was protecting the village

Shortly after Minato teleported the Nine Tails away from the village, he wanted to contact Hiruzen to inform him what had happened, but he was interrupted by Obito before he had the chance. From that moment on, Minato had his hands full with protecting Kushina, fighting Obito, and defending the village.
As shown later in the series, Minato's Flying Thunder God Technique has the capability to teleport others with him.... as long as he provides them his chakra to do so. Had he had the opportunity, Minato probably would taken a few shinobi with him to aid him.
With that said, since the village no longer in immediate danger from the Nine Tails, Hiruzen and a group of shinobi pursed Minato in attempts to aide him. Unfortunately, they finally arrived at Minato location after everything had taken place.

He attempted to reach the Third Hokage so that he could inform him of what happened, but was interrupted by Tobi
[...]
No longer being controlled, the Nine-Tails began retreating from the village, though it continued to attack the Konoha ninja that pursued it
[...]
Seeing the Third Hokage nearby, Minato communicated to him a final request
(Emphasis my own)