I was wondering if anyone has experimented with changing where they put the retention cord? I used to have issues with my accuracy if I changed fingers (like ring finger).

I personally always put it on my middle finger. Recently, (actually several years ago now) I changed the loop to be placed on the the middle phalange (between first and second joint of finger) but more toward the first joint  instead of all the way to the back of the finger (where a ring would sit). 

This change improved my consistency, (subjectively, no hard data). It also made it so that if I switch fingers, (but keep the placement the same, i.e. in the middle phalange) I don't seem to notice a difference in my shot placement. I think that it helps because it lengthens the lever arm from the wrist. It seems to add more control. I think it is akin to wrist snap when throwing. Has anyone else noticed anything like this?

I noticed Jaegoor places the retention cord in a similar position after studying some of his videos today. If anyone is struggling with accuracy and puts the retention cord at the back of the finger, this might be something to try.

I have a sling where I made the loop barely big enough to fit on my finger (on accident), so it is similar to a tightening one. I think I might switch to one that tightens in my future slings.

Yes. It affects the timing. The wider the gap between the retention and release cords is, the faster the ammo exits the pouch after you let go of the release cord. The timing difference will result in a small change in angle in the direction of the swing. It will be more obvious with a horizontal swing (like Byzantine) than vertical (like underhand).

If you consistently miss to one side after multiple throws, sometimes changing fingers is just enough to dial a shot in.

I have used everything from a wrist loop, (which pinched the nerves in my wrist and was quite painful) to a palm wrap, to almost every finger. 

I really haven't found much of a difference in accuracy.

Used to have the finger-loop on the middle phalanx of my middle finger for a long time, but now I have switched to the middle phalanx of my index finger as it feels to me more comfortable and more controllable. I never have the finger-loop to the base of the finger, as it feels less good for accuracy. Generally speaking though, I think something that has a much bigger effect on accuracy are release knots. Considering that all of the best 'shots' (that I have seen anyway) don't use knots, there must be merit to it. I personally can't stand them as I find they prevent a smooth and gradual release.

   I sling with the loop on the base of my ring finger.  I've tried other fingers, combinations of fingers, and a wrist loop.  I find using my ring finger the most comfortable and accurate.  Of course everyone is different.

I started with putting it on my index finger, but then I decided to put it around my middle & ring fingers together. I usually hold the down cord in my fist (lightly, with the knot against my little finger.) I've pretty much stuck with this ever since, though I'm not sure how it affects my accuracy.

after much practice i have also settled on the middle joint middle finger hold ..i sling underhand .i use a one turn method and as i bring my hand up to release  ago palm up and point at the target withe my middle finger ...For me it works like  a dream   I have Improved my accuracy by 70 percent ....of course that is not saying much.I use a 31 inch sling I am five foot seven tall
it workes like magic for me