Most information is from John Grigg's "1943". This book details the politics of World War II, specifically during 1943, in great detail. The book is highly critical of Churchill, but contains good answers to the question you have asked.
From the front of the book
Library of Congress Cataloging in Publication Data
Grigg, John. 1943, the victory that never was.
Bibliography: p. Includes index.
1. World War, 1939-1945--Diplomatic History. I. Title.
D748.G76 1980 940.54'012 79-22417
From page 175:
The conference was agreed to while at Quebec and the details were settled by a meeting in Moscow. This meeting was in August of 1943. The Tehran conference was in late November 1943. So there was a little over 2 months to figure out everything.
The first obstacle was getting everyone together.
Until a late stage there were doubts that the conference would ever take place. Teheran was as far outside Russia as Stalin would consent to go, but Roosevelt was reluctant to travel such a distance from Washington
Stalin tried to wriggle out of attending it himself, suggesting that Molotov go
President Roosevelt had his obligations as president, namely the signing of bills. It was only barely possible for bills to be transported to the area and back to the District of Columbia within the time period allotted by the Constitution of the United States.
Stalin simply did not want to leave Russia. After the victory in the civil war, Stalin maintained his power via cruelty and his cult of personality. For example, the critic Trotsky was executed at Stalin's orders after he fled the Soviet Union. Thanks to Khrushchev's secret speech given after Stalin's death, the political climate in the Soviet Union finally allowed criticism of Stalin's methods. Stalin did not want to attend the conference, because there was a very real chance that he would no longer be in power after returning to the Soviet Union. By using force to maintain his power, it required the near constant presence of Stalin himself to maintain his position. Although attempting a coup while he was present would be suicide, it would be possible while Stalin was absent. When you take into consideration what a cruel leader Stalin was, it is easy to gain backing for a coup. It would be hard for a coup to result in someone in power that was actually worse than Stalin.
I selected one passage from the book to support the notion of Stalin's barbarity, although many exist in just this text. From page 33
1942 was the only year of his wartime premiership when Churchill's
position seemed less than secure. Despite his immense personal
reputation, he was more vulnerable than either of his two partners in
the exclusive club known as the Big Three. Roosevelt, after the 1940
Presidential election, was virtually irremovable except by
death--until the completion of his unprecedented third term. And
Stalin, quite simply, was irremovable except by death, having no
elections to bother about and having either killed all possible rivals
or reduced them to a state of quivering terror.
I can find no evidence that Churchill objected to Teheran as the conference venue.
Churchill had traveled to Cairo on the HMS Renown from Plymouth. This ship was actually a World War I battlecruiser, but saw no combat according to Wikipedia. This ship was likely one of the more secure methods for Chuchill to reach Cairo. It certainly would be more secure from air attack than a civilian boat, although it was still vulnerable to German U-Boats.
It is more likely that security was achieved for Churchill by spreading disinformation of his whereabouts and through the use of numerous doubles. This isn't from the book, and is mostly speculation of my part.
Roosevelt traveled on the legendary USS Iowa to Cairo. This was one of the largest battleships to ever see combat.
Roosevelt and Churchill traveled to Teheran via air. They took of from Cairo. This would have been relatively safe at the time, given that the German forces had been expelled from North Africa. It is again likely that numerous decoys and disinformation would have been used to achieve security. History leaves us no concrete information as to the Luftwaffe's ability in the airspace at the time. But a lone transport would have been an appealing target. It was likely simply mixed in with various other routing flights from Cairo to Tehran.
From "Spearhead for the Blitzkrieg", the following quote on page 97
Once the Allies had built up their air forces, it became impossible for
the Luftwaffe to establish air supremacy
On the next page
To protect army units from air attack, the available fighters were
shared among the major army formations. Usually a fighter Geschwader
with three or four Gruppen operated within the command zone of each
group. It came under the command of the air fleet or air corps
headquarters responsible for air operations in that zone. During
friendly or enemy offensives, or if the air situation became critical,
these forces sometimes received temporarily assigned reinforcements.
After 1943, however, this was usually only possible at the expense of
other sectors on the front.
The "air forces" of the United States at the time were just the Army. There was no separate USAF during World War II. Operation Torch had allowed the ground forces to push out the German forces of North Africa in late 1942.
This source indicates that the Luftwaffe was barely able to operate in sufficient capacity to support the German ground forces by 1943. Even with good intelligence, conducting a mission to intercept the flight of a high profile target like Churchill would have been difficult.
From the front of that book, no cataloging data provided.
Once in Tehran, actual security was easy. From page 178 of "1943"
Persia was, like Egypt, nominally independent, but in fact under
foreign control. The twenty-two-year-old Shah had been brought to the
throne in 1941, when his pro-German father had been forced to
abdicate. The country was then occupied by British and Russian troops,
and later American troops as well.
The army from each respective delegation was present in the country already. They would have conducted preliminary assessments of the security of the air, as well as provided intelligence to the normal security details of each delegate.
Furthermore, specific consideration was given to Roosevelt's security
Churchill stayed at the British legation and Stalin at the Russian
embassy, which were close together. But the American legation, where
Roosevelt was at first accommodated--having declined the Shah's offer
of a palace--was about a mile away. Stalin then suggested that
Roosevelt move to the Russian embassy, for the sake of security, and
after some hesitation Roosevelt agreed.
Roosevelt actually stayed at the Russian embassy so he could be nearby and for security reasons. Given the attitude of the US & USSR in the Cold War towards one another, this seems almost like a fantasy. But Stalin deeply needed the military and materiel support of the Allies. Regardless of any political or ideological differences, it was in his best interest to be friendly toward Roosevelt. Any possible harm coming to Roosevelt by any source would jeopardize the American support for the Soviet Union. Without the steady supplies coming from the US, the Eastern front would have been much more difficult battle to win.
Of course I must note that on the same page can be found this quote:
For Stalin it was an early triumph, not least in that it put Roosevelt under twenty-four-hour surveillance by the N.K.V.D.
The NKVD was the police of the Communist party at the time. Roosevelt an his entourage were not fools. They likely restricted there conversations at the time to matter's pertaining to the conference only.