Don’t have any plans. Can someone see if there is anything amiss? Not sure how the block is attached to the ship. Should it just be hooked on so it can be released when lowering and raising the anchor?
Tackle'. There shouldn't be a rope attached to a tackle block (unless the rope in the image is obstructed from the view). From the outer end of the cat-head hung the catblock, usually having the same number of shaves as the cathead, and was suspended from it by the cat rope, which was lead through alternated shaves on the block and head, and back to the forecastle deck. 
Correct!
The cat stopper also fastens the anchor on. The cat stopper has not been used on all ships. The purpose of the cathead is to provide both a heavy enough beam to support the massive weight of the anchor and to hold the metal anchor away from the wooden side of the ship to prevent damage when the anchor is being raised from the water.
see above my response.
It is not necessary, not on all catheads you can see the use of stoppers. If the cathead has a special shave (on the side) for the cat stopper, then it is an indication that the cat stopper was used. Contrary, if you want to show the cat stopper, then you will need to make a stopper shave (see image few posts above)
1. The first question: the cat stopper end goes thru the anchor ring and tight to the timberhead.
The length of line should be long enough to allow the stopper to lower the anchor until the cathead tackle takes the weight of it. If you are depicting the anchor as fully secured into its storage place, the excess line can be formed into a hank or coil and hung by a loop over the bit, similar to how you hang hanks of line on a belaying pin. If you want the line to look as if crew are activity working the anchor, either preparing to lower it or weigh it, you can just leave the extra line coiled on the deck.
