In answer to your questions
1. I'm not at all sure that the F8F is slower than the Corsair, but, in any event, weight is not of great importance in maximum speed. A Tiger Moth is rather lighter than either the F8F or an F4U, but also somewhat slower. Power(and the efficiency of converting that power into propulsive energy) and the aircrafts drag largely determine max speed. And, of course, whether the aircraft has been optimised for max speed or for some other parameter - such as climb.
2. The Fw 190 set the standard in terms of efficient packaging of radial engines. Typical early American and English radial installations were somewhat crude by comparison (which is odd considering the American NACA cowling and British Townend ring were early steps in getting this right). The F8F and Tempest II/ Sea Fury both show signs of lessons learned from the 190. The British in any event did not, for much of the war, have radials that were particularly suitable for fighter use, since the whole British radial programme hinged around Bristol sleeve valve engines which matched extraordinary sophistication in their valve drive arrangements with superchargers that were straight from the Stone Age - and accordingly poor performance at altitude. The Japanese were also early adopters of well designed, closely cowled radials (occasionally with fan cooling) with individual, rear swept exhaust stacks to maximise residual thrust.
3. Theoretically there is no limit to the number of prop blades through which a synchronised gun installtion can fire, but, the more blades there are, the more the rate of fire of the gun is reduced. There were several Japanese fighters with 4-blade props and syncrohronised guns (Ki 84 and the NIKI- J Shiden come to mind as mass-produced examples. I'm sure there were more.)
4. Gyro Gunsights: I'm assuming that your question relates to WW2 Gyro gunsights - before transistors, integrated circuits or digital computers were invented. The basic principle is to use a gyroscope (which resists movement) to hold back the aiming point seen by the pilot in his gunsight so that, if the pilot aims directly at the target then the mean point of impact of the bullets will be ahead of that point. The challenge is to engineer it so that it can cater for different rates of turn, dirrecnt speeds, different ranges etc. In the early Gyro sights this was achieved by using an electrically driven gyro - spinning at around 4000 rpm - to which was attached a mirror that was used to project the graticule that the pilot saw in his sight. The pilot had to set the sight to the wingspan of the target, his speed and the altitude (since bullet drop is strongly affected by air density) and then, using a control on the throttle lever (which his hand would be on anyway) activate the sight as he turned with the enemy aircraft. As he turned with the enemy, within a second or so, the gyro and some very simple electronics would hold back his line of sight to the correct degree. This, the Mark II GGS, was also produced under license in the US as the K-14(USAAF) and Mk18 (USN).
Hope this helps.
NiallC