How are you using the rap ring? Do you just feed the tail through the ring and pull it through to choke the tree/ limb?
Also how are you rappelling? That CRIC looks pretty sweet. Do you have to take it off the rope when it's time to come down or am I missing something?
I don't use a carabiner on the end of the rope. I tie the paracord close to end of the rope so it's inline with the rope. It goes right through the rap ring. See an example in the video below.
I guess you could use one of those micro nite ize biners https://niteize.com/microlink-carabiner As long as the biner can pass through the rap ring your good to go.
The CRIC comes off the rope just like any other ascender. I take it off as soon as I get to hunting height. If the placement of the climbing rope allows me to use it as a tether, I'll use it, otherwise I switch to a tether once I'm on my platform. When I'm ready to come down, I connect to the F4, remove the tether and rappel.
I always carry a tether even I don't use it. If I get in some sticky situation while climbing or rappelling, I can use the tether to take the weight off the rope and fix whatever needs to be fixed.