Does every believer get an assigned responsibility in the future kingdom?
Yes. According to the context, every believer receives an area of responsibility in the future kingdom. Resurrected saints will have places of authority, and God will assign each believer responsibility, with rewards varying in amounts, as illustrated by the account of the talents and ruling over cities.
Les explains that the citizens who fill the earth during the kingdom will be normal flesh-and-blood people who have children, work, and produce, but in the absence of sin, the adversary, and under the perfect righteous rule of Christ the King. Resurrected saints will not be part of that flesh-and-blood population; instead, they will be interfacing with them from their place of authority. Les teaches that God is going to give every believer an area of responsibility, though beyond that he cannot explain it. He connects this to the account of the talents: the servant who received ten talents and gained ten more was told, when I come into the kingdom, you shall rule over ten cities. So Les likes to think the believer's rewards will be various amounts of responsibility in that kingdom economy. How it will be done, he does not know, but that is what believers are working for reward.