If you want to have good tap pressure, than a pump is required, if you only want to hand water the garden and there is enough head pressure in the tank then you may not need a pump, but in most cases a pump is the better option.
You will more than likely have a small leak somewhere, if the pump is connected to the internals of the home. The first place to check is the toilet, in most case the washer in the toilet system is leaking letting water in, place a piece of toilet paper in the back of the toilet pan, if the paper gets wet, you have a leak.
The air pressure system allows for a draw off of water before the pump will start, depending on the size of the pressure tank will determine how much water may be drawn off before the pump will start. An pressure tank will only turn off to high pressure, this means if your water supply runs out, you need to manually turn the pump off.
The constant pressure system turns on almost immediately when you turn a tap on and will give constant pressure hence the name. This type of electronic controller will also shut power to the pump should you run out of water, this is called Run Dry Protection.
A pressure tank can also be connected to a constant pressure system pump, and be used as an energy saving devise, when sized correctly the pump will not turn on for small draw offs of water.
Yes all pumps should be protected from the elements, it is important to protect the motor from water entering it. Most pump manufactures do cover warranty for water entering the motor or electrics. Pumps should not only be covered adequately but they should be positioned so that ground water pooling does not enter the pump housing as well. A pump cover will only extend the life of you pump, be sure the cover has adequate ventilation vent installed.
Yes, because these pumps normally are not used very much, only in an emergency. As a starting point any unused fuel should be drained from the pump as fuel will go stale. The pumps should be started very couple of months and run for a few minutes, check all outlets for blockages and broken parts. If you have any problems starting the pump or running smooth then a service should be arranged.