My answer would be no. This would not be possible.
Voldemort is evidently the most powerful dark wizard in the world. When he ruled, there was fear amongst everyone, terror across the world.
Every week, news comes of more deaths, more disappearances, more torturing ... The Ministry of Magic’s in disarray, they don’t know what to do, they’re trying to keep everything hidden from the Muggles, but meanwhile, Muggles are dying too. Terror everywhere ... panic ... confusion ... that’s how it used to be. Harry Potter and the Goblet of Fire
Severus against Voldemort, he doesn't stand a chance. Dumbledore on the other hand, is the only wizard that Voldemort ever feared. Dumbledore could have dueled for an exceptionally long time, but remember what Dumbledore said himself?
"You flatter me," said Dumbledore calmly. "Voldemort had powers I will never have."
This said, Voldemort may not be the better wizard, but in fact have Dark Magic that Dumbledore wouldn't dare to use. Using this magic may have made their powers on par. Even then, Voldemort was still weak at that time.
I have strength enough for this.
Voldemort in Philosophers Stone.
As to an inescapable room, I highly doubt that would be possible. Azkaban has had many breakouts, but is still probably one of the best and secure jails. The most the Ministry would possibly do is to make a new "jail", specifically for Voldemort, due to his past.
You could keep giving Quirrell food, but as per spookypeanut answer, I believe that Voldemort would possess Quirrell not to eat it. If Quirrell dies, Voldemort would transform into a misty, ghost like substance, and escape would be easier.
You also said, "the whole set of problems that occur further along in the story would have been easily averted." Out of universe, the whole entire series revolves around Harry and Voldemort. Even if he was captured, there would be no story whatsoever.
Also, remember this is Voldemort that we are talking about. It may be Quirrell's body, but he had the Wrath of Voldemort behind him this entire time. To try to bring Voldemort to justice, to a jail cell, is the equivalent of Earth shifting it's course away from the sun.