The input I am looking at is the 5-pin type (https://www.adafruit.com/product/1699).
That should be fine, but you only need 3 of the pins. The extra pins are so you can pass-through the signal to a pair of speakers and switch-off the speakers when headphones are plugged-in.
1. Is there a way to use both Left and Right channels from the jack? Say, to average them somehow before sending data to the Arduino?
You can run both signals through resistors (about 10K). Connect other resistor-ends together and to the Arduino input. This will give you a mixed (summed) signal without affecting the original L & R audio signals.
Another option is to use 2 analog inputs (with 2 bias circuits) and mix in software. It's just a matter of summing. Analog mixers are built-around summing amplifiers and digital mixing is done by summing (sample-by-sample).
2. Looking in this post (https://forum.arduino.cc/index.php?topic=476900.0) I've seen two different methods of processing the signal from the jack before sending the the A0 pin, one an envelope follower, and the other a sort of biasing circuit.
Use the bias circuit. There are 2 issues with the voltage follower. You loose all of the frequency information (which makes FFT useless), and there is a voltage drop of about 0.7V across the diode which means if the signal is low, nothing will get through. You can use an active voltage follower (an op-amp circuit) to solve the voltage drop, but you still loose the frequency information.
And, with the bias circuit, either leave-out or increase the value of R1 . A 1K load is OK for headphones but too low for a line-level signal, Plus, when used with the above "mixing resistors" all of these resistors make voltage divider which will reduce the signal, so I'd just leave it out, especially if you mix the left & right signals.
FYI - There is a chip called the MSGEQ7. It gives you 7 bands filtered and then output as a DC voltage (like a voltage follower). That makes your software a LOT easier if you can live with 7-bands. That chip doesn't require bias on the signal.