Can someone please explain me what exactly is going on? which patch is broken and in which way?
I currently have version 1.06 installed, do I want to update or not?

Version 1.06 has a problem in one of the missions and cannot be completed. GOG's version 1.07 fixes said problem, but completes the game after you finish any mission. There were some files to patch GOG's 1.07 to a working one, but not sure if they are still up.

I think the last working version was 1.04. So if you still have a backup copy of that installer, you shouldn't delete it.

As far as I know 1.07 also works fine as long as you have installed the The Japanes Incentive DLC.

1.04:
Completely working version.

1.06:
Georgia Avenue is broken. You can't finish that map

1.07 4 DLC:
Game breaking bug which causes the ending credits to start when a map finishes. This goes for both a new game and for saves from an earlier version.
Georgia Avenue is still broken.
The 4 DLC maps are dead as there are no cars and civilians in the streets.

1.07 4 DLC with Japanese Incentive:
Installing the JI DLC on top of the 4 DLC version fixes the ending credits bug.
Georgia Avenue is still broken.
The 4 DLC maps are dead as there are no cars and civilians in the streets.

1.07 Gold Edition:
This is equal to the 4 DLC plus the JI DLC. Only difference is that you don't have to do a seperate installation for the JI DLC.
Georgia Avenue is still broken.
As I don't have this version, I have no idea whether or not the 4 DLC maps are dead.

Apparently Kalypso is working on a 1.08 version which should be the final version. However they've also stated on their forums that they don't have the people to actively support and old(er) game.
Both GOG.com and Kalypso are aware of the problems.