Good, basic questions
You're OK, Jason
Use a steel screw-link (Maillon Rapide), which are rated. I like the Demi Rond, and am using it currently. Before that, I used the Delta which worked really well for me, and I'll probably go back to that some day. This is not a good place to use a caribiner
Use the Maillon to attach your <i>right side</i> D-ring to the thimble on your steel core flipline. This is permanent. You'll be adjusting with your left hand. Right side gets busy with chainsaws, slings, etc.
On your left-side D-ring will clip the Snap if you're fully extended, or the micrograb for adjustment.
While ascending your rope, your flipline should go around your mids twice, counterclockwise. If you're not too fat, you should be able to clip the snap to your left-side D and it'll be out of your way for active climbing.
Letting the flipline just hang gives the chainsaw the opportunity to contact the flipline (not good), as well as getting in the way of what your feet are doing.
You say you are clipping the snap to the D on your left, and using the ascender on the thimbled end for adjustment on your right? Technically, that'll work, but ascenders are not recommended for steel core fliplines, and they're not easily adjusted with one hand, like the micrograb is. You're so close, only a screw-link away! You can definitely streamline the system. Stay safe.