Why do we have to inflate our tires to a set psi every once in a while?
Inflating our tyres of cars turn the shape of tyres to more cylendrical. So because of this you will not get any anti torque by the normal force that would oppose your torque provided by the engine. That is why you might have seen that it is difficult to drive with a flat tyre.
As we drive the tire loses air? It's because of the difference in pressure?
As we drive the tyres heat up and thus increasing the pressure and further increasing the pressure difference.
Yeah as the pressure difference arise gases have a tendency to leak and will obviously leak even as you can never turn it completely leak proof as air can even diffuse.
A proper installed new tire, will lose pressure over time? or we have to pump air due to poor installation?
A proper installed tyre will lose air but obviously after a long durtion rather than a worn out tyre. This is because of the thinning of the rubber tyre surface or due to the punctures that you got repaired(as isaid before air would leak)