The Fury shared no components with the Tempesr V, and the Sea Hawk was totally different. .
Quoted from the History of the Sea Hawk on "Thunder and Lightnings"
here.
While Hawker Aircraft Ltd. was introducing their Sea Fury prop-driven naval fighter in 1944, they began to consider how to apply the new jet-engine technology to the Fury design. Rolls-Royce had just introduced a new engine, the B.41 (to be named the Nene), that had adequate thrust to propel the big Fury airframe, and in November of 1944, Hawker provided a preliminary concept of a B.41-powered Fury to the British government, with the designation P.1035.
The design team, working under Sir Sydney Camm, removed the Centaurus radial engine from the nose of the Fury, moved the bubble canopy as far forward as possible, and put the Rolls engine in the middle of the aircraft, with intakes alongside the fuselage and exhaust as a tailpipe going straight out the back. After a little more thought, the Hawker engineers came up with a follow-on proposal, designated the P.1040, in December, 1944; the new design had a major refinement in that the straight tailpipe was replaced with a split tailpipe, with each outlet in the wing root; this scheme meant the wing root had to be thickened, and so the intakes were placed in the wing roots as well. This layout reduced the amount of ducting and allowed fuel to be stored both fore and aft of the engine, preserving the aircraft's center of gravity as fuel was consumed. Avoiding having a long tailpipe also meant avoiding any thrust loss effects which were keenly felt on the earlier lower powered jet engine designs.
P.1040 VP401 lifting off on her first flight; unknown origin
The Fury's distinctive elliptical wing was abandoned for one with straight edges (to simplify manufacturing) and the tailplane was raised to allow it to clear the jet exhaust. The new design also had tricycle landing gear - the first Hawker aircraft to do so; it no longer looked very much like a Fury. Armament was specified as four 20-mm Hispano Mark 5 cannon. The P.1040 was intended for RAF use. Government interest, however, was mild; the war would clearly be over before long, and both the RAF and the Royal Navy had other jet fighter designs either flying or in the works. Nonetheless, despite the end of the war, in October 1945 Camm ordered the construction of a prototype.
By this time, RAF interest had disappeared completely - their air planners believed that their latest Meteors would be the hottest thing in the air for the foreseeable future - and Hawker Aircraft, faced with massive cancellations of orders for their prop fighters, hastily modified the P.1040 design for carrier operation and submitted the proposal to naval planners in January, 1946.
I won't bother answering all of the other points but I could back up all of my other statements if required.