Why it didn't auto download? How to install it?

These optional DLCs never auto-install on GOG. You have to manually "buy" it for free and then install it either via the website or the Galaxy client, whichever variant you are using/preferring.

I finished the purchase just a minute ago but it doesn't install and it's not on the DLC list how do you install this?

edit: ok the problem is with the Galaxy client. I can see and download the DLC from the web page but it doesn't show anywhere on the Galaxy client. Haven't tried to install it yet I hope I can at least manually install it
edit 2: manually download and install from the web page, galaxy is just f**ked up. Don't have time to play and test if it's there though, it doesn't show up on your DLC list on the galaxy client even after you install with the installer you get from the web page.

This is actually a known problem, this is a functioning workaround:

https://www.gog.com/forum/pillars_of_eternity_series/rum_runners_package/post8

They're in the available DLC list at the right of the game info window, within GOG Galaxy now. Pressing Buy and then Install works.

Be warned though, despite being just a couple of megabytes they can take a long while to download and install with the method the galaxy client checks your current installation. Took me ~11 minutes to install the rum runners and the other free dlc even though the game is installed on my SSD.

It should also be noted you HAVE to run the game through GOG Galaxy to even use these things. Even after they are installed (some authentication nonsense?)

If I run the game via a steam shortcut the DLC popup will MENTION being installed, but the icon (lower left of the main menu window) is greyed out. In game you cannot pick for example, the drunk personality.

Running the game through Galaxy, the icon is lit and the personality can be selected.

Sucks. I was streaming this game to my couch via steam in home streaming, but now all GOG dlc is going to prevent that?

I guess I can try adding gog galaxy to steam or something.

currently there's no DLC list visible on my galaxy client (there used to be something like (4 DLCs), eventhough I have captains footlocker etc. Backups and goodies page doesn't show the rum runners as well and the so called fix won't work for me since I manually downloaded and installed it with GOG's own installer already.

Actually their installer scans your computer, locates the game folder and handles the installation with ease. If only galaxy had that same simple feature they used since day 1 on their installers :D

I guess I'll just stop using galaxy for now it's easier and more convenient that way.