It could have been tested in a very simple way: sit the bomb on the ground (or in a tower), and set it off by remote control. The first atomic bomb test (Trinity) was done in this way, with the bomb in a 30m high tower:
The first hydrogen bomb test was similar, with the 74 ton bomb, "Mike", being essentially a building:
However, a ground-level test like this results in a lot of fallout - the ground below the blast is made radioactive by the absorption of neutrons from the blast, and is vaporised and sucked up by the fireball, and falls out of the sky later (thus, "fallout"). The Soviets did two things to reduce fallout from the Tsar Bomba test:
The bomb was detonated at a height of about 4km above the test site. While the fireball was large enough to reach the ground, the shock wave from the blast, reflected by the ground, stopped the fireball from reaching the ground.
The bomb tested was a half-power version of the full design. Hydrogen bombs use a fission bomb as a detonator, which triggers a nuclear fusion reaction in the surrounding material (lithium deuteride is the usual choice, with both the lithium and deuterium (heavy hydrogen) contributing to the fusion reaction - the first US test of a lithium deuteride bomb (the Castle Bravo test) had a yield 2.5 times greater than predicted, since the prediction assumed no contribution from the lithium). The efficiency of the fusion reaction can be increased by surrounding the fusion stage by a tamper that will reflect neutrons back into the fusion stage. For a high-yield weapon, uranium-238 is used for the tamper: not only does this reflect neutrons back to increase the yield from the fusion stage, the neutrons will trigger a fission reaction in the uranium-238. This results in a high-fallout bomb due to the radioactive fission products. The Tsar Bomba, as tested, used a lead tamper, which is estimated to have halved the yield. With the full high-yield fission-fusion-fission design, the yield would have been about 100Mt, at the cost of a lot of fallout.
A 4km high test made things more difficult for the Soviets. The bomb was dropped by a modified Tu-95 Bear, from an altitude of about 10km. The fall of the bomb was slowed by a parachute to give the bomber (and the accompanying observation aricraft) more time to escape before the bomb exploded. The safety of the bomber crew was not guaranteed! In the event, they survived, after the pilot regained control of the plane after the loss of 1km of altitude after the blast reached the plane (the plane was a little over 100km away from the test site when the blast caught up to it).
The bomb was tested over Novaya Zemlya, which was a regularly-used test range. The plane took off from the Kola Peninsula
The bomb was somewhat bigger than really feasible - at 27 tons, it was at the limit of what the Tu-95 could carry, and the weight meant range would be reduced. In terms of the area destroyed by the explosion, big bombs are not efficient - the same amount of fusion fuel can be used to make multiple smaller bombs, which can be spread over the target to destroy a larger area than the big bomb. The multiple smaller bombs are also easier to deliver.
As for cost, the Tsar Bomba was basically a giant version of existing already-tested designs. This minimised development costs. The bomb itself would have been more expensive than a smaller bomb, since it used more material. It would have had a similar yield per kg of fusion fuel used, and would have cost about the same as, or somewhat less than, 5 10Mt bombs.
For some further discussion by u/restricteddata on the Tsar Bomba, see:
For video of the explosion, see:
The buildings being destroyed are probably in the village of Severny, 55km from ground zero - every building in the village, brick included, was destroyed.
For more video of the test, including the weapon being dropped, see:
See:
for photographs of the bomb and the weapon being dropped.
