When to use which semiconductor?
Silicon has many advantages that have made it the dominant semiconductor material: 


A native oxide. This is key to the development of the MOSFET.
Relatively good physical robustness. Some other competing materials are more fragile, leading to losses in production simply due to mechanical breakage of wafers.
Abundance in nature. Silicon is the 2nd most abundant element in the Earth's crust, making it easy to mine, though refining it to the purity needed for electronics is still a significant effort.


Further, since silicon is so widely used, economies of scale make it much cheaper to produce chips or devices in silicon than in other semiconductors.

So, if silicon will do the job, we will almost always choose silicon to achieve low cost.

We might choose other materials if we need


A direct-gap material, typically for optical sources like LEDs.
A specific bandgap. For example for photodiodes for detecting 1550 nm wavelength, a bandgap less than about 0.8 eV is needed.
High carrier mobility, which allows higher frequency devices. For this you'll see materials like SiGe, GaAs, GaN, or InP used.
A specific lattice constant, for growing one material epitaxially on a substrate of another material. The ability to engineer both lattice constant and bandgap is why you see ternary and quaternary compounds like GaAlAsP used.


I'll leave aside the question of how dopants are chosen because 1) I know almost nothing about it, and 2) the choice of dopants is likely different for each semiconductor material.