1. Right idea, wrong units of measure that leads to errors and does not account for system losses. You work with Watt Hours, not Amp Hours. 12 volts x 15 AH x 5 hours = 900 Watt Hours.
2. Battery Capacity = 5 days x Daily Watt Hours / Battery Voltage. 4500 WH / 12 volts = 375 AH battery
3 You might have troubles that may or may not work out for you. Depends on your location. 600 watts of panels should work unless you are in the Pacific NW. With 6 panels you will wire them 3S2P and you must use a MPPT Controller. Your calculations are incorrect to to size controller. I will give you the formula and let you figure out what is wrong. MPPT Controller Current = Panel Wattage / Battery Nominal Voltage. 620 watts / 12 Volts = ?? Amps. Do you have an issue?
4. Common mistake, hope it works. A 1000 watt Inverter @ 12 volts will take 80 to 90 AMPS at full power. A 12 volt 150 AH battery can only deliver 15 to 20 amps, enough for a 200 to 250 watt inverter.
5. Again right idea but way off on numbers. This is because you do not understand units of power and basic ohms law. A 1600 watt generator @ 13.3 amps only tells you the generator voltage = 1600 watts / 13.3 amps = 120 volts AC. 1600 watts @ 12 volts DC = 133 amps, enough to blow up a 150 AH battery in 1 minute. The Generator will work, all it needs is a 50 amp 12 volts DC charger. 12 volts x 50 amps = 600 watts just like your panels, or about 30% of your generator capacity.
Keep digging, you are on the right path. I will help you, but you must do the work yourself and find the answers. When done you will know how and why. Right now I can tell you a 75 or 150 AH is grossly undersized, and with 620 watts of panels will cook them with 50 amps of charge current. With 600 Watts of panels you are looking at a minimum 300 AH battery with 375 being a great fit based on your numbers. You may not like the results, but at least you will know why and the consequences if you try.

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