1) It is not a "hot water heater", it is a water heater (it makes hot water).
2) Gas is not more economical to operate. An electric water heater is nearly 100% efficient while the heating element is on. There is no loss of heat up a chimney as with a gas unit. In idle mode both water heaters are nearly identical for loss of heat through the insulation. However, the cost per BTU to heat the water is usually more for electric than natural gas. Hence, it may cost more to heat water with electric than gas but it depends on where you live.
3) A residential water heater with dual elements does not use both heating elements at the same time. The upper thermostat is designed for when the upper part of the tank is cold, it will turn on the upper heater element to heat water so as it is draw off the top, it will be at least warm water at the faucet. When the water reaches the set temperature of the thermostat it switches the power to the lower element. The lower element will turn off buy the lower thermostat when it reaches temperature.
4) All electric heaters are the same. No! depending on the model or size, check the wattage rating of the heater elements. Most are in the 3800 watt range but some can be 5500 watt. Never replace a 3800 watt unit with a 5500 watt unit unless your electrical connection can handle the load. For the average house the water heater is wired with 10 gauge wire on a thirty amp circuit. Older homes have 12 gauge on a 20 amp circuit.
5) A gas unit can heat the water faster than electric (called heat recovery rate). Not necessarily. It all depends on the model itself; electric wattage ratings or the size of the gas burner. Each water heater will have this rating and it is not how fast the water will reach temperature of the tank, but how fast it can rise a given amount of water. It takes the same energy to raise 40 degree cold water ten degrees as it does to raise 75 degree cold water ten degrees. So if you have an electric 30, 40 or 50 gallon tank with a 3800 watt element, each will raise the water temperature at the same rate, the difference is when will the entire tank would be hot enough to use say for a shower.
On #2 I'm not quite sure how you're divorcing "economical" from "cost per BTU". Economics is about $/BTU x efficiency, not just efficiency.
On #3, hell YES it will operate on both elements, at least some of the time, even if they're controlled separately. (Perhaps you meant that to read "A residential water heater with dual elements does not always use both heating elements at the same time?)
On #5, do the math. Only a few electric tanks will put out a max of over 4500W, which is 15,354 BTU/hr, at an efficiency of ~100%, delivering 15, 354 to the water. The 5500W electric heaters deliver 18,766 BTU/hr.
Atmospheric fired tanks have a steady-state efficiency of ~80%, and very few have burners with input BTUs smaller than 30,000 BTU/hr, which delivers (0.8 x 30,000=) 24,000BTU/hr to the water. That is a heat rate over 25% higher than a 5.5KW electric, and 60% higher than a 4.5KW electric.
Most 40 & 50 gallon gas-fired tanks these days have burners north of 35KBTU/hr, and recover in typically half the time of comparably sized electric tanks.
I don't know of ANY electric tanks that will recover as fast as the smallest-burner gas fired tanks, but maybe bein' in the biz and all you can point to some specific models where that would be the case?