excuse me saying, but it's not a boiler and it's not a tank, it's a hot water cylinder.
I think from the colour that you are in Ireland and I can't tell how old it is and how thick the insulation (there is colour coding).
If you have the opportunity to heat it with a gas boiler, it will probably take around half an hour (depending on boiler, cylinder and plumbing method). If you have a timer, set it to come on half an hour before you get up in the morning, and half an hour before you come home in the evening. This will give you a full cylinder. You can set the timer to go off at times when you do not expect to need much hot water, which will save a small amount due to waste heat loss from the pipes.
It is probably around 100 litres, and using the electric immersion heater takes about 100 minutes (but an immersion heater may not heat the entire tank).
A gas boiler typicaly has between four and ten times the power of an electric immersion heater, but cylinders may not be able to absorb heat so fast, especially if old.
Where I am, electricity cost me around 16p per kWh, and gas around 4p per kWh, so it costs about four times as much to heat the cylinder using electricity as using gas. Have a look at your bills or tariff to see what you pay.
For economy, use the gas boiler, and turn off the electric immersion heater except for those days when the boiler breaks down and is awaiting repair.
As you have the square cold-water tank above the cylinder, it will tend to be at around room temperature, and is open to the air, so yes, it is possible to get infected and support bacterial growth. If you use plenty of water it will tend to get flushed out, but you can't expect it to be sterile. if it has a close-fitting lid, that will reduce risk of dust, dirt, spiders and mice falling into it.