Hmm so the observation is that actually Germany uses its air on the UK already, and the best way to take Cairo early is by doing an airstike of the full luftwaffe. Based on the idea that you need offensive land troops to overpower Russian I will use the following builds.
Just a single investment (7 IPC) in getting Cairo, the bombers have the option to attack Cairo in G4 or be used against Russia instead.
G1: 6 Artillery, 1 Submarine
G2: 10 Tanks, 1 Bomber
G3: 1 Transport, 4 Bombers
I1: 1 Tank, 1 Mech
I2: 2 Tanks, 1 Mech
I3: 2 Bombers
Your German fleet will try to save to Bismarck with a strafe and get to S91 in G2 hopefully taking Morocco. Then to S93 to join with the transport and pick reinforcements of lost units in Morocco. In G4 you attack Egypt with your small fleet (2 transports) and Luftwaffe that is 7 bombers, 6 Fighters and 5 Tacticals minus losses from UK naval battles.
Taking Morocco for Italy in G2 combined with swarming the med with the Luftwaffe will quickly rise Italy to 30 IPC per turn. With Egypt defeated this rises even more to 40-50.
Japan will force the USA to focus on them with JDOW1 and pearl harbour