I'm going to attempt to answer several of your questions pertaining to your fire pit courtyard/sitting wall, and paver patio. The patio was installed first and compacted, it was restrained around the edge of the patio with (I'm guessing) brick paver edging with 10" spikes. Because the curve is pretty tight, I'm hoping they installed the spikes every 8-10" in the provided spacer holes.
They then layed out the bottom course of block for the fire pit and marked a line around the outside of the fire pit block. Then they removed the bottom layer of block and cut the individual bricks. After all cuts were made they were left with a perfect circle. I'm hoping they removed the sand in that area and added another 1/2 - 3/4" of base stone and compacted by hand with a hand compactor. Then, they (hopefully layed in two small pipes for screed rails within the circle and covered with sand and screeded of the sand and removed excess sand.
They then installed the first course of small retaining wall block, making sure to level it from front to back, and side to side. After the 1st course of block was perfectly level then they brushed off the block and ran (2) beads or lines of block adhesive and started installing the 2nd course of block making sure to stagger the joints (possible 1-2 block may have been cut to allow this to happen. This process was completed until the desired courses/height of block had been installed. Then the wrought iron fire pit insert was installed, this is extremely important because the steel fire pit rings keep the r.w. block from getting too hot-for they are not fire proof.
As for installing your firepit and patio area over a former area where a tree once stood-I would be very apprehensive. I would locate it to another area if possible, but if another area is not available thenthis is what I would do for the installation. You could dig it out by hand, but you may not be able to lift and/or move the stump even if you get it dug out. I personally would hire someone with a mini-excavator to come in and excavate the tree stump. The excavator will be able to chase the roots, as well as dig pretty deep if need be. They can also lift a reasonable amount of weight, carry the stump out to the street, and place it in a dump truck, trailer, etc.
Whatever you decide to do to get rid of the stump, do not place the removed soil back in the hole. You will never get it adequately compacted so that after you patio has been installed and you start to get a little rain, it doesn't sink. I would place filter fabric and/or geo-textile in the bottom of the hole (after the existing sub-grade has been compacted) and layer with a 3/4" clean, angular stone or limestone and compact every 3-4". Do not exceed 4" in stone without first compacting, the average plate compactor can handle max. 4".
After you have brought your hole back up to grade with the stone and finished your compaction. Grade the surrounding area, lay another layer of filter fabric and/or geo-textile and commence with installig your layers of stone base for the entire patio, compacting every 3-4" layer. We automatically excavate every patio 9" deep from the top of the finished brick patio elevation. A 9" excavation allows us approx. a 6" depth of compacted 3/4" angular base stone and 1" depth of 2NS Sand. The pavers are generally 2 1/2" tall.
I also recommend you run stringlines at your proposed finished brick elevation and pitch the string if need be. You can buy 2' pieces of re-rod to use as grade stakes at Lowes or Home Depot, and we use an average of 10 to as many as 20 depending on the project. We also try to use recycled concrete that is equivelant to 23A crushed limestone with fines (crushed concrete is recycled and less expensive than the limestone - at least in our area. The crushed concrete will need to have fines in it, this will make for a more stabil patio and you will achieve proper compaction. I also highly recommend that you lightly sprinkle each layer of stone with water, this will also assist in proper compaction. Do not saturate it so that the water is pooling, but get the stone good and wet so that is barely mushy. You would be amazed at how much more compacted your area will become.
If you are starting to think this seems like a lot to do for a simple patio. If you have had any brick paver installers bid your project, this is why the good ones charge the amount that they do. Just like anything in life, if you do not properly prepare your foundation, your project will fail or develop problems over time. Good luck and I hope everyone out there agrees with my directions. I wish you the best with you fire pit/patio project!