I second Rachel's comments. Tight-fitting rings are key to avoid skin slip. I usually size rings such that there is a 3 mm to 4 mm (max) gap between the shell and the inside edge of the crown ring. That leaves just enough space to accommodate the rope loops.
Once you have the loops on the crown ring, you should be able to just push it over the bearing edge and down the side of the bowl with some force. If you have to help it along with a few taps with a mallet, that's still OK.
If the crown ring is so tight with the loops on that you can't get it to come down over the bearing edge, it needs resizing. (Too small a crown ring will crack the shell, or leave grooves in the side of the shell where the rope is pressed into the wood, or the ring will open up at the weld point because there is too much tension on it, or you will tear the skin where it is squeezed between the shell and the loops.)
The flesh ring should be the same size as the crown ring, or maybe fractionally smaller. If the rings are on the loose side, use the smaller one as the crown ring. If they are on the tight side, use the larger one as the crown ring. Ring size must not differ by more than 1/3 the thickness of the rings. If the crown ring is larger than this relative to the flesh ring, you run the risk of pulling the crown ring down over the flesh ring on one side during the dry pull.
The other key to avoiding skin slip is the number of knots. The skin is held in place only in the spots where the rope presses down onto the skin against the flesh ring. In between the cow hitches, nothing holds the skin in place at all. So, more knots means more pressure points, which makes skin slip less likely. 28 - 32 knots is a reasonable range.
If you have a rolled steel (smooth) flesh ring, make sure to wrap it generously with cloth to a thickness of about 0.5-1 mm of fabric. This allows the skin to deform itself around the rope where it contacts the flesh ring, which helps to lock it in. If you wrap the flesh ring too thinly, that deforming doesn't happen, and the skin is more likely to slip.
If you have a flesh ring made out of rebar, either don't wrap it at all (works just fine), or wrap it generously. A measly layer of wrap removes the friction you get from a ring that's not wrapped at all, and it disallows the deformation (dimples) you get in the skin with lots of wrap. In other words, a thin layer of wrap on a rebar flesh ring is worse than thick wrap and worse than no wrap at all.
For rebar flesh rings, I no longer bother with the wrap. It simply isn't necessary. For rolled steel rings, I wrap generously.
Another trick is to use the first of the hitches shown here
(the one using black rope). This hitch doubles the number of pressure points where the skin is trapped, making skin slip basically impossible. The double-strand hitch does the same thing, and you may like the chunky look. Be aware that this version is difficult to tie evenly and consistently, and that tying it off cleanly is difficult too. (There are detailed explanations in my book for how to do this.)
To keep a floating bottom ring centered, the trick is to apply tension by hand evenly and slowly. Go around three to four times by hand, tightening a little bit more each time until, on the final round, you are pull as hard as you can by hand. Also, when starting to tighten on one side, tighten three or four verticals, then go to the opposite side, tightening three or four verticals there. Then to the same thing at 90 degree angles, so the ring is held in place by the higher tension at the four compass points around the perimeter. When tightening with a bar during the wet pull, check the ring position every few verticals and, if it looks like going off-center, tighten on the opposite side to re-center the ring before continuing.
If you end up with a lop-sided bottom ring after the wet pull, no problem. Let the skin dry, then loosen the verticals a little, re-center the ring, and take out slack very gradually over several rounds to keep it centered.