It was about as similar to the UK Parliament as a Republican system which believed in separation of powers could be. Both are bicameral legislatures, with representation from given states in the house of representatives being equivalent to the commons. The Senate is an "upper house" similar to the Lords in that it is not all that undemocratic (states being represented equally not by population) and actually is similar to the idea of "Knights of the Shire" where in the British parliament two knights from each county would be nominated to sit in the commons (Davies, R.G.; Denton, J.H.; Roskell, J.S. (1981). The English Parliament in the Middle Ages), alongside other delegates. Two Senators from each state is demonstrably a derivation of this.

J. A. Chafetz, Democracy's Privileged Few: Legislative Privilege and Democratic Norms in the British and American Constitutions (Yale, 2007) is worth a look.

So they really didn't reject everything about the Westminster system. However, they did ditch the hereditary and monarchical aspects. As well as attempting to put in place a separation of powers, because the British Prime Minister and Cabinet act traditionally variously as individuals drawn from the legislature, forming the executive and one member of which was head of the judiciary... The US thought this was too much power in too few hands and so distributed power according to the checks and balances principle of the legislature and judiciary being independent of the executive and so reducing the likelihood of tyranny.

R.A. Goldwin, Separation of Powers - does it Still Work? from 1986, has a lot of your basic groundwork on why the US system is how it is, (the first chapter being almost entirely devoted to your query re the US system) in the early chapters, then highlights factors which he cited as erosion of the principle over time.

I would add that the Framers were very apprehensive about the idea of "party government," which they saw as detrimental to good policy and an erosion of the idea of separation of powers. To that end, they designed a system that (in their minds) would limit the effects of parties and factions. It didn't take very long for parties to arise naturally, though, and as you noted the Constitutional system inadvertently makes a two-party duopoly somewhat inevitable.

See Richard Hofstadt, Idea of a Party System

Very interesting, informative responses, but the reason the US is ruled by a two-party duopoly is because of the electoral system, not because of the structure of the government itself.

Elections in the US are based on the First Past the Post (FTPT) system, wherein the candidate who wins a plurality of votes is the winner. The UK and [edit: as commenters below note, the UK system is FTPT as well, and UK Parliament is largely a two-party system as a result] many other parliamentary democracies use one of several types of proportional systems, where all candidates who receive votes receive some representation in the legislature.

Example: in the US, a candidate for the House of Representatives who wins 50.1% of the votes (or 33.1% in the case of a three-way race, etc.) will simply win the seat. Whereas in many parliamentary democracies, candidates winning 50.1% of the votes can win 50.1% of the seats in the legislature.

The result is that elections in the US often put a single party in power as the overall winner, and generally only one other party has the wherewithal and the mass appeal to challenge the ruling party on a regular basis: hence: duopoly. In a parliamentary/proportional style election, many different parties can win seats in the legislature. There is, however, a kicker, and the kicker is this: in order to govern, all of the parties winning seats are forced to form a governing coalition. They have to ally with other parties in order to be able to pass legislation, take executive action, etc.

So in a way, the systems don't wind up being all that different in practice. In parliamentary systems, the "center-right" coalitions and the "center-left" coalitions tend to be dominant in any government, alternating control of the government every few years. In the US, the Republican Party is effectively itself a "center-right" coalition made up of multiple factions that might well have their own party and brand identity in a parliamentary system. Likewise with the "center-left" Democratic Party. The major US political parties are effectively coalition parties.

EDIT: The principle in political science explaining why FTPT and similar electoral systems tend to favor two dominant parties is known as Duverger's Law:

Duverger suggests two reasons this voting system favors a two-party system. One is the result of the "fusion" (or an alliance very much like fusion) of the weak parties, and the other is the "elimination" of weak parties by the voters, by which he means that voters gradually desert the weak parties on the grounds that they have no chance of winning.

A prominent restrictive feature unique to this system is purely statistical. Because the system gives only the winner in each district a seat, a party which consistently comes third in every district will not gain any seats in the legislature, even if it receives a significant proportion of the vote. This puts geographically thinly spread parties at a significant disadvantage. An example of this is the Liberal Democrats in the United Kingdom, whose proportion of seats in the legislature is significantly less than their proportion of the national vote. The Green Party of Canada is also a good example. The party received approximately 5% of the popular vote from 2004 to 2011 but had only won one seat (out of 308) in the House of Commons in the same span of time. Another example was seen in the 1992 U.S. presidential election, when Ross Perot's candidacy received zero electoral votes despite getting 19% of the popular vote. Gerrymandering is sometimes used to counteract such geographic difficulties in local politics but is controversial on a large scale. These numerical disadvantages can create an artificial limit on the level at which a third party can engage in the political process.

The second unique problem is both statistical and tactical. Duverger suggested an election in which 100,000 moderate voters and 80,000 radical voters are voting for a single official. If two moderate parties ran candidates and one radical candidate were to run, the radical candidate would win unless one of the moderate candidates gathered fewer than 20,000 votes. Observing this, moderate voters would be more likely to vote for the candidate most likely to gain more votes, with the goal of defeating the radical candidate. Either the two parties must merge, or one moderate party must fail, as the voters gravitate to the two strong parties, a trend Duverger called polarization.

To add to this question, was a presidential form of republican government in United States a new idea that was never attempted before or was it based on a government of another state? If it was based on older idea, what was the source of Presidential form of Government?

The framers also paid a lot of attention to the failed Roman Republic, in which (as was the understanding at the time) Caesar was able to use the republican form to overthrow the republic and take tyrannical power. If all power comes from the legislature, there are only a few men you have to corrupt to - as the theory goes - obtain all the power for yourself. So this, combined with their distrust of democracy (seriously!), and and combined with "powerful tendency in the Legislature to absorb all power into its vortex" (at least according to some framers, including Madison), led to a situation where the framers became huge proponents of separation of power and were never going to seriously entertain a system where the executive was subordinate to and dependent on the legislature.