The previous thoughts were really strong. I think that there is not one particular role because the framers themselves were divided into different camps. The Anti- Federalists saw the role of the federal government in a lesser light, a more non- interventionist approach being taken. In this light, the federal government was seen as almost adversarial, having the potential to take away individual rights and something from which defense was needed. This was a position that Jefferson and his Republican party adopted later. The opposite viewpoint was held by the Federalists, who viewed the federal government as a source of power and a source of greatness for the nation. Armed with the cosmopolitan belief that the best and the brightest should be in the position of federal power, the federal government was a realm where greatness in both legislation and execution could be present. I think that the vast difference in perception about the federal government can be attributed to the divergence in its embrace.
The Constitutional Convention was called because the Articles of Confederation had failed. The main problem with the Articles of Confederation was that they did not give the central government enough power. Under the Articles, congress did not have the power to tax, to raise troops for an army, to regulate interstate commerce, to settle disputes among the states, to stop states from issuing money, to control trade, to place tariffs on foreign goods and to enforce laws, among other things. Because of this the young United States was falling apart and in danger of failing. The Founders saw the role of the Federal government under the Constitution in two ways. First, the Constitution had to make the federal government strong enough to keep the country together. Therefore, it gave the federal government much more power than under the Articles of Confederation. Second, the Constitution had to keep the federal government from becoming too strong and overwhelming the states. Therefore, the founders incorporated the ideas of checks and balances, federalism, and separation of powers into the Constitution. During the debate over ratification, the founders saw a third role for the federal government under the Constitution and that was to guarantee the rights of the people. Therefore it was agreed that as soon as the Constitution was ratified, there would be a Bill of Rights added to it.
You can argue about this some because there were many different people at the Constitutional Convention and they had different ideas about what the government should be like. People like Alexander Hamilton had a very expansive vision for the federal government while others thought it should be much less powerful.
In general, though, the role the Framers foresaw was more limited than what we now have. The federal government was going to mainly provide two things: defense for the country and a good business climate.
The first of these is obvious, the second less so. What I'm saying is that the Framers wanted to have a strong economy and they thought the national government should make that happen. They thought that it should prevent trade wars between states and that it should maintain a single currency. Many of them hoped it would create infrastructure that would help the country grow.
So, in general, the government was supposed to be a supportive, rather than an instrusive, entity. It was supposed to keep Americans safe and help them have a chance at prosperity.
The answer to that question would depend on who you asked, as the agreement they came to was not universal among them. Alexander Hamilton for example was an ardent Federalist who believed in a strong national defense, wanted to create a national navy with direct federal involvement in the economy. He created our first national tax system as Secretary of the Treasury and some say he wouldn't have minded if George Washington had been appointed King.
Thomas Jefferson, on the other hand, an anti-federalist, saw a much more limited authority for the government, and he and Madison supported the Bill of Rights which placed many of those limits. Patrick Henry was terrified of any federal government at all, and opposed giving them the authority to tax or raise an army.
In general, the mainstream of the convention delegates wanted to have a government that could pay the nation's bills, provide some stability economically, and guard the frontier and the coast against potential enemies.