Good questions. For the answers we shall go straight to the Tao Te Ching:
Chapter 8: "...In conflict, be fair and generous. In governing, don't try to control..."
Chapter 17: "When the Master [of Taoism] governs, the people are hardly aware that he exists. Next best is a leader who is loved. Next, one who is feared. The worst is one who is despised. If you don't trust the people, you make them untrustworthy. The Master doesn't talk, he acts. When his work is done, the people say 'Amazing: we did it, all by ourselves!' "
Chapter18: "When the country falls into chaos, patriotism appears."
Chapter 31: "Weapons are the tools of violence; all decent men detest them. Weapons are the tools of fear; a decent man will avoid them except in the direst necessity, and, if compelled, will use them with the utmost restraint. Peace is his highest value."
Chapter 46: "When a country is in harmony with the Tao, the factories make trucks and tractors. When a country goes against the Tao, warheads are stockpiled outside the cities."
Chapter 49: "The Master has no mind of his own. He works with the mind of the people."
Chapter 57: "If you want to be a great leader, you must learn to follow the Tao. Stop trying to control. Let go of fixed plans and concepts, and the world will govern itself. The more prohibitions you have, the less virtuous people will be. The more weapons you have, the less secure people will be. The more subsidies you have, the less self-reliant people will be. Therefore the Master says: I let go of the law, and people become honest. I let go of economics, and people become prosperous. I let go of religion, and people become serene. I let go of all desire for the common good, and the good becomes common as grass."
Chapter 58: "If a country is governed with tolerance, the people are comfortable and honest. If a country is governed with repression, the people are depressed and crafty. When the will to power is in charge, the higher the ideas, the lower the results. Try to make people happy, and you lay the groundwork for misery. Try to make people moral, and you lay the groundwork for vice. Thus the Master is content to serve as an example and not to impose her will. She is pointed, but doesn't pierce. Straightforward, but supple. Radiant, but easy on the eyes."
Chapter 59: "For governing a country well, there is nothing better than moderation. The mark of a moderate man is frreedom from his own ideas. Tolerant like the sky, all-pervading like sunlight, firm like a mountain, supple like a tree in the win, he has no destination in view and makes use of everything life happens to bring his way. Nothing is impossible for him because he has let go, he can care for the people's welfare as a mother cares for her child."
Chapter 60: "Governing a large country is like frying a small fish. You spoil it with too much poking..."
Chapter 61: "When a country obtains great power, it becomes like the sea: all streams run downward into it. The more powerful it grows, the greater the need for humility. Humility means trusting the Tao, thus never needing to be defensive...If a nation is centered in the Tao, if it nourishes its own people, and doesn't meddle in the affairs of others, it will be a light to all nations in the world."
Chapter 66: "All streams folow to the sea because it is lower than they are. Humility gives it its power. If you want to govern the people, you must place yourself below them. If you want to lead people, you must learn how to follow them."
Chapter 68: "The best leader follows the will of the people."
Chapter 75: "When taxes are too high, people will go hungry. When the government is too intrusive, people will lose their spirit. Act for the people's benefit. Trust them; leave them alone."
Therefore, in direct response to your questions: Yes, Taoists do care about the role of government in their lives. About a fifth of the Tao Te Ching is about leadership and the role of the government. They want a government that lets people be, that trys to make the nation PART of the world instead of something that goes AGAINST the world. They want a government that will have a policy of noninvolvement and moderation in rule. The Taoist view of government can be said to adhere to their religious principles, but this does not mean that they want a government that involves itself in religion. Most Taoists hold that there are MANY available paths to spiritual well-being other than just Taoism, so they would never expect their government to dictate the religion of the people, nor to involve itself in religious matters. "Just be" is a very Taoist principle, and Taoists want the government to "just be" and to allow the people to "just be".