Hey guys, I read the documentation (and went through several videos on Youtube) and set the skeleton just like described. All bones except for root and hips (pelvis) have translation retargeting set to Skeleton. Root and hips have Animation scaled option set. The base poses for retargetting were very similar.
Thank you MKenner for the information, it was helpful but this eventually did not end up as a retargeting problem - I mean the twisted finger, not the normal map artifacts - these probably require a different solution.
Finally I found the error to be where I least expected it - being the animation from Mannequin folder in TPP Template itself: ThirdPersonRun.
Bones middle_02_l and middle_03_l have their axes’ orientation different from all the other fingers.
This cannot be seen very well when using the manneqin mesh as the mannequin’s fingers are divided to segments and each segment has its own bone - but you can see that they actually are flipped upside down when viewing from close distance. This can also be seen in the following image.
I exported the animation from Unreal, imported it to Blender (this process however created an animation with 16 frames instead of original 18, I do not know if this is an error or not, but the starting and ending frame seemed the same as in the original animation, so I continued), I changed the orientation of bones middle_02_l and middle_03_l to be the same as in the rest of the fingers and copied their pose to all the animation frames. I eventually ended up with what can be seen in the image below.
You can also find the exported animation in the following link - it is not perfect but it is the correction I was looking for (you can import it to UE4 by assigning Mannequin skeleton to it upon import).
[ThirdPersonRunCor][3]