Sources at bottom.

To answer your second question: Sometimes. Nixon knew about bigger operations, such as coup attempts, plots etc. The easiest example for this is the approval of the plot to overthrow Castro that eventually led to the Bay of Pigs. Nixon was a part of the discussion during that meeting (March 17, 1960), and actually had to point out a legal issue with the way Eisenhower wanted to proceed (long-story-short, Eisenhower thought the OAS would support intervention based on Castro being a dictator, Nixon pointed out the OAS technically only supported intervention if international communism was involved). But yes, Nixon was aware of much of the CIA's activities.

Answering your third question: No. Allen Dulles controlled the CIA. Whether Foster may have controlled Allen at times or not is arguable, and the degree to which Allen attempted to exercise power out of sight of Eisenhower is also debatable. However, Dwight Eisenhower loved covert interventions, and put a great deal of faith in the CIA's ability to secretly intervene (Iran, Guatemala, Indonesia, Congo, +all the planned or proposed interventions), and took a very involved hand in all aspects of foreign affairs. So between Eisenhower, the Dulles brothers, C.D. Jackson, and various other State and Defense officials involved in foreign affairs, Nixon's role was definitely important (especially as concerns resistance and criticism of the administration's policies), but he certainly was not in control of the CIA.

As to your main question, that's a lovely question that I'm not able to answer sufficiently in this format. However, I would argue that his experiences as VP certainly contributed to the attitude which led to Watergate. Eisenhower hated leaks, especially when they reflected poorly on him. He once directed Douglas Dillon Robert Murphy to tell Christian Herter (Secretary of State), to keep his damned mouth shut, after a leak. What's more, Nixon learned as VP the incredible power the President could wield by controlling access to himself and, through his "friendship" with J. Edgar Hoover, the seeming ease with which places could be bugged - gaining valuable information.

But his belief that he could "get away" with Watergate is another issue entirely. Nixon as VP was quite a different animal compared to President Nixon. As VP, Nixon was beholden to patrons (mostly JFD but also Eisenhower), and was regularly forced into roles he disliked because he lacked leverage, power, or control. There's no clear and undeniable link from Vice President to President Nixon, but it doesn't seem far fetched to posit that his experiences as VP contributed to shape him into the kind of President who would attempt such a fiasco.

Sources:

Nixon as part of 5412 Committee and restraining Eisenhower Memcon, March 17, 1960, FRUS: 1958-1960, Volume VI, Cuba, 861.

Telling Herter to keep his mouth shut - Telephone Conversation, Eisenhower to Murphy, July 15, 1958, 1, Telephone Calls July 1959, Box 43, Diary Series, Ann Whitman File, Dwight D. Eisenhower Library.

For a different perspective than the one I and Benjamin Goldberg have on the Nixon-Dulles, Nixon-Eisenhower dynamic, see Irwin Gellman's The President and the Apprentice: Eisenhower and Nixon, 1952-1961 (New Haven: Yale University Press, 2015) and Jeffrey Frank, Ike and Dick: Portrait of a Strange Political Marriage (New York: Simon and Schuster, 2013).