For the first part of your question, no, JFK did not ask NASA if it was possible to go to the Moon. JFK was TOLD that it was possible, after he made broad inquiries about how could America surpass the Soviets after the USSR's early space success of Yuri Gagarin's April 12th, 1961 Vostok 1 flight (in which Gagarin became the first man in space and scored a MASSIVE propaganda victory for the USSR).

The first inklings of a potential lunar program for America were presented to JFK on April 28th, 1961 (two weeks after Gagarin's successful Vostok 1 orbital flight and one week before Alan Shepard's successful Freedom 7 sub-orbital flight). It was at that time that JFK was presented with a short internal memorandum by Vice President Lyndon Johnson that was titled "Evaluation of Space Program". In part, the document stated that it was believed that the first "dramatic" space achievement in which the USA could be expected to surpass the USSR in the Space Race would be by pushing for a manned lunar landing (the document also stated that it was believed the goal could be achieved, assuming strong Presidential, Congressional, and public taxpayer support, by as early as 1966 or 1967, and certainly by the end of the decade).

It is well appreciated that Kennedy was not really an overt fan of the space program or NASA. He was however a consummate politician, and he was made to recognize the huge political propaganda capital that could be gained by supporting NASA and the civilian space program, and also was made aware of the tremendous advantage that dominance of space could afford, in not just the civilian, but also both the much more covert military and intel arenas. Those propaganda victories could be achieved by setting a lofty longer-term target that would really challenge the Soviets and allow America to eventually leap ahead of the USSR's early successes of Sputnik and Vostok/Gagarin - with that long term goal being to land a man on the Moon and return them safely to the Earth before the USSR could.

Lyndon Johnson, Kennedy's Vice President at the time, was actually one of the driving forces behind the idea of implementing a plan to send a man to the Moon and returning them safely to Earth. It was he - VP Johnson - who, immediately after Alan Shepard's successful Friday, May 5th, 1961 sub-orbital flight in the Mercury Redstone-3-boosted Freedom 7 spacecraft capsule (in which Shepard became the first American in space during his brief 15-minute flight), called together a group of powerful military and political insiders for a meeting. This meeting was led by VP Johnson, and included powerful guys like Robert McNamara (then the Secretary of Defense) and James Webb (then the current head administrator of NASA). This at-the-time classified Top Secret meeting took place the weekend after Shepard's successful flight, on Sunday the 7th of May, 1961. At that time, the Vice President told the military and NASA reps he wanted them to come up with a comprehensive and persuasive rationale for a Moon mission - an argument that could be used to persuade President Kennedy to give the go-ahead for what ultimately came to be known as the "Apollo Program".

To directly quote parts of the then-classified document that came from this meeting - "It is man, not merely machines, in space that captures the imagination of the world...Dramatic achievements in space therefore symbolize the technological power and organizing capacity of a nation...major achievements in space contribute to national prestige. This nation [America] needs to make a positive decision to pursue space projects aimed at enhancing national prestige...The non-military, non-commercial, non-scientific, but "civilian" projects such as lunar and planetary exploration are, in this sense, part of the battle along the fluid front of the Cold War".

The next day, on Monday, May 8th, 1961, this classified document was presented to President John F. Kennedy by Vice President Johnson, literally just minutes after President Kennedy had very publicly awarded America's first astronaut, Alan Shepard, with the Distinguished Service Medal for his Mercury Freedom 7 flight that had taken place just a few days earlier. Two days later, on May 10th, 1961, Kennedy, after consulting with a slew of senior military, budgetary, science and civilian advisors - heavy hitters like NASA head James Webb, SecDef Robert McNamara, National Security Advisor McGeorge Bundy, Doctor Hugh Dryden, Ed Welsh (who was the Exec. Secretary of the National Aeronautics and Space Council). That meeting was an important one, because it was during that discussion that it was decided by Kennedy to ratify the recommendations he had been presented with, which in large part focused on giving the go-ahead for a lunar mission.

Again, President Kennedy did not come up with the idea of a lunar landing program. Politically, it was VP Lyndon Johnson who was actually the real driving force behind getting the ball rolling on what came to be called the "Apollo Program", and Johnson was the guy who (with a lot of support from NASA and the military) was the real driving force to convince JFK to back the idea.

As for the second part of your query related to the Saturn design - early incarnation designs of what would eventually become the mighty Saturn V booster were already on the drawing board at the time. These rocket designs were given the "C" designation, coded as "C-1" (which became the Saturn 1 booster) through to the final Saturn V variant (also originally known as the "C-5"). The three-stage Saturn V booster system was really the brainchild of German rocket scientist ex-pat Wernher Von Braun, who, with a bunch of help from the fellow rocket scientist gang at the Marshall Space Flight Center/Redstone Arsenal, was already, at the time of Alan Shepard's Freedom 7 flight, working on expanding on their military-centric Jupiter rocket booster program by developing what they called the Saturn/C-series booster system. Saturn was in the early stages of design and development (though not highly funded) even prior to JFK giving his famous May 25th, 1961 "'Special Message to the Congress on Urgent National Needs" address to a joint session of Congress in Washington DC, where he stated (among other things) that "I believe that this nation should commit itself to achieving the goal, before this decade is out, of landing a man on the Moon and returning him safely to the Earth." A few months later in September of that year, Kennedy also gave his famous "We chose to go to the Moon..." speech at Rice University in Houston, Texas, which signified the ramping up of the public call for support for an American lunar program. It was really on January 10th, 1962 however that NASA publicly announced plans to fund and create the mighty Saturn V (C-5) booster needed for the planned Apollo lunar program, but early design and R&D work had begun on working out the basic requirements of the C-5 Saturn booster system much earlier than that.