First question, what saddle are you using? That may change your tether setup a little.
1.
Most folks use some variety of 11mm static line for tethers and lineman belts. I think Yale Bandit is a popular rope, I just checked tree stuff and didn't see it. I really like the Sterling HTP but I've got the 9mm and am not sure if the 11mm is that much different. I think I started with 15 feet before knots and I feel like I need a longer tether, my next one will be 20 feet before I start, but...I like a higher tether than most. I got my 11mm static from REI and was able to get it by the foot, but the rope sucks, it is really stiff and doesn't do knots very well. I am seriously thinking about ordering one from Treestuff.com and having them put sewn eyes on both ends of a tether to reduce bulk, we'll see it's 15/sewn eye so I'm gonna have to think about it, lol.
2.
I would get a bunch of the screw lock carabiners. I have both auto and screw lock, and they both work but you can forget to screw one shut. I use one on my ropeman which is at the bottom of my tether and also the end of the rope is tied into that same carabiner, this allows me to adjust my tether length, which is useful if you use an XOP hand climber or for just different size trees. I use a purisk knot for my saddle, so a carabiner there as well. I should note that I use a sitdrag and a rock climbing harness. So I'm always tied into my tether at least once, usually twice.
3.
I have 2 ropemans, I use one for my lineman belt and one for my tether. I bought a ropeman 1 and a ropeman 2, I wish I had only gotten the ropeman 1, the 2 has teeth and will work on smaller rope but it almost works too well and doesn't release very well and also it tears up the rope a little more than the ropeman 1, in hindsight I would just get 2 ropeman 1's. Use one on the bottom of your tether and one on your linemans belt to adjust it. You will need a carabiner for each one as well, so at least 4 total if my math is right, lol.
Oh and for your tether the part that goes around the tree will typically be a figure eight know or overhand with a bite, that you girth hitch around the tree, so no carabiners needed there. I've seen some folks use them there for going around branches quicker, so maybe something you want to play with, I never did that, so I can't speak to it, maybe someone else can chime in.
Alright hopefully that helps, good luck, let me know if that doesn't make sense or you have other questions, I just looked none of my pictures show my whole tether, lol.
Ropeman 1
https://www.amazon.com/Wild-Country...8422386&sr=8-1-spons&keywords=ropeman+1&psc=1