mAirList is designed to be used with a mixing console. A common setup would look like this:
First, you need a PC with multiple sound cards, or a single multichannel sound card offering 3 or 4 stereo outputs (e.g. a 5.1 card) - two outputs for player A and B, and one for the cart players all of which play on the same sound card. In mAirList’s configuration software, set the audio device configuration accordingly. (Note that mAirList 3.0 supports multichannel sound cards only in the paid versions.)
Each stereo output is connected to a separate fader on the console, so you have three faders “player A”, “player B”, “cart”. The mixing console is used to set the volume of the players as needed. This is why the main players keep their volume when you start a cart player - the DJ is supposed to that on the console. After all, it’s live assist 
The players A and B are used to broadcast the music files one by one (which was loaded into the playlist). If your desk offers remote start (all radio desks and some DJ desks do), you can even wire it to the PC (there are multiple options available, e.g. using a USB gamepad) and make mAirList start the players instantly as you open the faders on the desk. No need for a mouse or keyboard anymore. Jingles, IDs, etc. can either be inserted into the playlist, or you can use the cartwall to play them on the third fader.
Cueing is done by using the “PFL” feature. There are two ways to perform PFL: You can either load the item into a player and start that player in PFL mode (clicking the small “headphones” icon in the bottom left corner), or you can double click the item to bring up its Properties dialog - there’s an embedded player on the last tab page, sometimes referred to as the “PFL player” or “Extra PFL player”. As you already noticed, clicking a playlist item’s icon brings up the Properties dialog and starts PFL instantly. That’s just a shortcut, which can be disabled in the config.
The PFL output of all players, including the embedded PFL player, can be routed freely among the available sound cards. Have a look at the “audio devices” page in the configuration software, it’s pretty straightforward.
During PFL playback, the “PFL cue dialog” pops up which is use to set the various cue markers (cue in, cue out, fade out, etc.). If you want to skip part of a file, say the intro, this is the place to do it. Once an item is on air, there is no way to skip audio - this is something you would never do in a live situation anyway.
Automation mode is meant for unattended operation, although you can still click the Next button to skip a song manually. The automation system will start and stop your players just like a DJ would do manually. If you want the songs to be crossfaded, mAirList will only do so if the “start next” or “fade out” marker is set on the items. You can either do so manually, or you can enable the auto cue feature in the config (Misc -> File Import) and let mAirList set the marker automatically according to some dB threshold.
There is no way to go backwards in the playlist, but all played and deleted items are moved to the Recycle Bin which can be accesed from the browser, so you can easily bring back an item which you deleted accidently. You can also set the “number of items to keep in the history” in the configuration to, say, three, which means that a buffer of the last three items will be kept at the top of the playlist, displayed in gray and with a trash can symbol - for reference, but also to restore them easily if needed (click “Recycle” in the context menu).
This is how all major stations are operated - at least here Europe. I heard that in the US, you rather have some sort of “automation” mode without separate faders even in live situations - is that true?
