(As to your last question, see my previous post about Norman Mineta, who was a former Democratic member of the House of Representatives.)
The primary difference between the US Presidential system and most parliamentary systems is strict separation between the executive and legislative functions of government. In a parliamentary system, whichever party or coalition can command a majority of the legislature gets to choose the head of the government, and that person gets to choose other legislators from his party (or coalition) to serve in important positions. This means they have both an executive and legislative role, simultaneously.
The people who wrote the US Constitution were wary of the power wielded by a unitary parliament that was easily manipulated by a monarch (as Britain’s parliament was in the 18th century.) So they designed a system where the executive branch was entirely separate. To add a check to Presidential power, they required that the Senate confirm choices for important Presidential appointees. (And it’s the Congress as a whole, through writing enabling statutes for government agencies, which gets to decide who does and does not require Senate approval to appoint, with the exception of some positions, such as Supreme Court justices, which are mentioned in the Constitution itslef.)
Because of this structure, it’s common to have situations where the head of the government (the President) is of a different party than the legislature (the Congress) which would be impossible in a parliamentary system. Right now, both houses of Congress have Democratic majorities while the President is a Republican. During the Clinton administration, the House and Senate both had a solid Republican majority for six out of his eight years in office.
Finally, unlike a parliamentary system, there is no requirement that the Cabinet positions be filled by legislators. Cabinet appointees have been former Senators and Representatives, former Governors, outstanding civil service employees from within the department in question, and frequently, private citizens who may have previously left politics or never even been politicians at all. So the party affiliation of the appointee may not be a contentious issue, as long as the person agrees with the President on the policy issues for the portfolio assigned to him.