1) Square is fine, most of the time people are using what they have for material and tools. A round firebox is easier to make than a square one.
2) You should have enough draft with the 3, 3/4" holes and the 2" bung at the top. Just make sure your intake is below your outlet. You may want to find a way to control the outlet. Some people use magnets, other use a flap of metal that covers the whole hole. What ever you use, be careful, it will be hot.
3)If you do a search for the "Minion Method" here you will find different ways of doing it, but they all accomplish the same thing. A slow controlled burn of the charcoal. Hardwood lump will work for this, it is what I use. Start about a 3/4 full chimney of charcoal. If you have an old clean metal coffee can, with bottom removed, you can put that in the middle of the firebox and pour your unlit charcoal around it. When the lit charcoal is ready, pour that in the can and then remove the can. This will allow the lit coals to start the unlit coals next to them in an even fashion.
4) You will have to experiment with your UDS on this one. I would start with the whole 8# bag and see how long it goes. If you have to you can always shut down the inlets and outlet to smother the fire and use the remaining charcoal on the next cook. I add the wood dry and spread around the firebox. Put on close to the lit coals so it will ignite right away. If you are doing large pieces of meat you will not need as much smoke at the end of the cook as you will need at the beginning.
5) This is another area where you will have to experiment. I would start with everything wide open. Then close off your capped inlets first, one at a time, and see what the effect is. Every smoker is different. IMHO, I would wait 20 min. to see what happens. The Brinkman you use did not have the control of air that a UDS has. You will find that will make a big difference.
6) I really don't have an opinion on this one. You may want to spray with Pam before your first cook and that will help.
7) I use a bottom of an old Brinkman Gourmet smoker. The thing about an ash catcher is that it needs to allow air to circulate through the bottom of the firebox. It is not critical but helpful. Your ash catcher will have to be square also. I would be on the lookout for an ash vacuum. These are all metal and will not catch on fire if there is a stray lit coal.
I use the small vent hole in the lid. Drill out a cork that will fit in the hole and stick the thermometer of your choice. I use one like the ThermoWorks-Original Oven Thermometer.
For $15 that is a good deal. If you want to get the probe closer to grate level, order the Probe Holder/Clip. You would be able to drill a hole in the side at grate level and use the clip to hold it in place.
As for your first cook. I would go with a pork butt. Reason being, if you have wild swings in temp. they are very forgiving. Also, they are cheaper than brisket, so if something does go wrong you won't be out that much $$.
I hope this helps. I am sure others will have other opinions on this, but this is just what I have learned.
Oh and don't forget to take pictures. A lot of us can't read and like to look at the pretty colors.