From what I understand, I will have to take the tree out of its pot, lay it on the ground, chop a fair amount of roots (this part would need some clarifications) and then put it back in a pot.
That's pretty much it. I generally start a repot by cutting the bottom half of the root mass off, but why don't you remove the bottom 1/3. I use a pruning saw with changeable blades. Actually, I have 2 saws - 1 for actual pruning, the other for root work. When my rootwork blade gets dull, I move the blade on the pruning saw to the root saw and a new one goes on the pruning saw. Onct that's done, I bare-root the tree. I try to work in shade, out of the wind, and I am VERY careful to ensure the roots remain wet the entire time they are exposed (not in moist soil).
So, my questions would be:
• How do I decide which roots to cut?
Once the bare rooting is complete, I remove problem roots. These are roots NOT attached to the base of the tree. I remove crossing roots, roots growing upward or straight down, roots growing back toward the center of the root mass. If roots are j-shaped or circling, I prune them back so they are straight. In all, you can remove about 1/2 - 2/3 of the root mass.
• What kind of soil should I in the pot after? Would an all-purpose mix work?
I'm not sure you have the luxury of the time it might take to put together the kind of soil that allows you to water to beyond the point of soil saturation, so you're flushing the soil as you water, without any need to be concerned the soil will remain saturated so long it seriously limits root function and potentially wrecks root health. If you can find pine bark (often sold as mulch or clay soil conditioner or just soil conditioner) you can make a medium that looks like this:
If you can't find it quickly, it really is a goal worth working toward if you're serious about keeping plants healthy over the long term. I haven't used anything commercially prepared in well over 20 years. It's a snap to make, less than half the price, and the results are as different as night and day - no comparison to media based on all find components.
• Do I put the tree back in its original pot or do I use a larger one? The current pot is nursing pot of 10-inch. The only bigger pot I have is 13-inch terracotta pot.
If you happen to find the bark, the pot can go back into the original pot or a smaller one. If you can't find the bark, do this:
Find some items to use as ballast.
These pots are full of soil and the shaded area is 100% saturated soil The ht of that soil saturation will be the same after every thorough watering. Note how the bricks displace soil that would otherwise be saturated? If you added 2 bricks to the pot on the right and turned the bricks on their edge, you will have eliminated about 90% of the soil in that planting that would be capable of holding excess water. The idea is to displace as much of the soil in that shaded area as possible with ballast. You can use empty soda bottles with the cap on if you like. The only proviso is, there has to be an unbroken soil column from the top of the soil to the bottom of the pot. Ballast is a passive way to eliminate excess water, and it works great. It will allow a grower to use media (s)he would otherwise have to fight tooth and nail against for control of the plant's vitality. I can explain the science or link you to something that does if you're interested.
So, once the ballast is in place, mix up enough soil made of 1 part perlite to 1 part potting soil and fill in the bottom 6" of the pot. The ballast shouldn't be taller than 6". Set the plant on top of the bottom layer of soil, the n fill in the rest of the way with a mix of 2 parts potting soil:1 part perlite. Use a piece of dowel rod or a thin piece of wood to work the soil into the roots as you back-fill around the roots. Then, moisten the medium.
• How should I deal with the water during the process? I use a tub or a hose with a mist fitting on the end
Fogg-It fitting.
You should plan on doing this outdoors. If you have everything ready to go, it should take about 2 hours the first time you do a repot. I find it helpful to secure the tree so it can't move in relation to the pot. This prevents breakage of hair roots from even slight jostling as the tree is reestablishing.</i></b>
Al
These pots are full of soil and the shaded area is 100% saturated soil The ht of that soil saturation will be the same after every thorough watering. Note how the bricks displace soil that would otherwise be saturated? If you added 2 bricks to the pot on the right and turned the bricks on their edge, you will have eliminated about 90% of the soil in that planting that would be capable of holding excess water. The idea is to displace as much of the soil in that shaded area as possible with ballast. You can use empty soda bottles with the cap on if you like. The only proviso is, there has to be an unbroken soil column from the top of the soil to the bottom of the pot. Ballast is a passive way to eliminate excess water, and it works great. It will allow a grower to use media (s)he would otherwise have to fight tooth and nail against for control of the plant's vitality. I can explain the science or link you to something that does if you're interested. 
Fogg-It fitting.
