By Andrew D Bird, this account of the activities of the Banff Wing and their Mosquitos will answer most of your questions.
The Mosquitos used two warheads. The 25lb solid head was used for sinking all kinds of ships, and were generally fired hoping for half the hits above water and the others below. The 60lb HE warhead was used for attacking the defending flak ships and land targets.
The nature of the attacks, generally into narrow fjords, meant that they were unable to avoid flak despite the stand-off nature of the weapon. Near the end of the war there were several attacks on U-boats attempting the transfer to Norway. The 6pdt Molins gun (57mm) was also used successfully against U-boats and small vessels.
Remember that although the rockets may be fired out of effective range, the aircraft's flightpath would mean it had to fly into more dangerous volumes of sky after release.
The 5" rocket may have been better in that respect, but this saw insignificant, if any, use in Europe.
The USN found the 20mm ineffectual against Kamikazes, although I would suggest that against more conventional attacks its effectiveness would depend on achieving a hit and where - a 20mm into the engine would certainly spoil a Wildcat pilot's day, and as they would be viewed head-on in the attack that would be what mattered for a hit. Hits from larger weapons would always be more effective.
German Army flak notably the quad 20mm, appear to have been very effective against attacking aircraft. These would have been likely to have been mixed with the 37mm, so allocating successes woudl be difficult.
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British aircraft used 3.5-inch rockets.
The solid warheads were mainly intended for use against U-boats as their pressure hulls were very tough, but a solid warhead would usually penetrate leaving the U-boat unable to submerge.
The RN found even 4-inch shells could have difficulty penetrating pressure hulls. They produced special "shark" projectiles to help penetration.
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