Electronic – Why don’t people tend to use voltage dividers or zeners in front of linear regulators

Arduino Leonardo, like the Uno, uses an NCP1117 linear regulator (LDO), which can supply 800 mA, but at 11.1 V input that would be 4.88 W, so thermal protection will set in much earlier.

Using USB is a possible solution, but keep in mind that USB doesn't have to supply more than 100 mA, you'll have to negotiate for 500 mA. And Arduino is not designed for low-power, I don't know what the Leonardo consumes, but I guesstimate the Uno's supply current at 50 mA, so powered from USB that may leave too little for for instance a series of LEDs.

Your own switching power supply looks like the best solution. Bypass the NCP1117 and supply it directly with 5 V. A 12 V to 5 V buck converter should be able to have a near 90 % efficiency. If the dingus would need 50 mA it can run for more than 48 hours on the LiPo battery.

Further information
Arduino Leonardo schematic

Answers to 1 and 2 should come mostly from the datasheet of whatever switching power supply chip you plan to use. First you have to make sure the chip is intended for your input voltage, output voltage, and maximum current. After that you need to follow the directions, carefully.

These kinds of chips are designed assuming a certain range of parts external to them. The datasheet will tell you what is acceptable and it may even give you some guidance what values to use over a range of parameters. For example, the datasheet might recommend a larger inductor for the upper half of the input voltage range. However, you'll probably pick a chip because it's max characteristics exceed your requirements by some reasonable margin. In that case, you follow the recommendations for the max case, since that's effectively what you will be using. Some datasheets assume this and only give you one set of recommended values. Just follow them.

As for question 3, yes, switching power supplies can cause more EMI. The main trick is to keep all the local loop currents localized to the power supply sub-circuit, and to keep these loops physically small. This means starting with a good layout, then carefully routing the key nets. Make a separate local ground net for the switcher, then tie that to the main ground in only one place, probably right at the ground side of the output capacitor. This keeps the loop current thru the diode, inductor, and output cap localized and off your main ground plane. Only the net delivered current should flow thru the power and ground feed points from the power supply sub-section.