I like to use "pickling" salt and superfine sugar to make brines because each dissolves in cold water. (Both are just sugar and salt that are ground finer than normal.) Most big-box grocery stores carry these.
Otherwise, yes, you need to use warm or hot water to get the salt and sugar to dissolve. The issue with using warm/hot water is that you then need to wait a while to get the water temperature back down so you can add the meat.
There are two basic ways to deal with this:
1. The quick and simple method is to use some warm water from the sink and mix it vigorously with the salt and sugar. Give it time to dissolve. Then add the rest of the water for the bring, using cold water.
2. A second way to deal with this is to heat half of the water, mix it with the salt, sugar, spices, etc. When it dissolves, add it to an equivalent amount of ice. (For example, a gallon of water is about 8 1/3 pounds. So you can use a half gallon of hot water plus a little over 4 pounds of ice. Don't worry about getting the measurements that precise - it isn't that important. I certainly wouldn't weigh the ice. I would just eye ball what looks like a half gallon.) This will leave you with a cold brine and you can then add your meat. Also, some spices give off more flavor when added to warm or hot water.
I usually use a slightly stronger brine (i.e. more salt per given amount of water) than Jim's, but it is best to start with a weaker brine since it gives you more room for error.
To cook, first cook directly over the heat for about 2 minutes per side (or until nicely browned).
Then, as you noted, use an indirect heat zone with the lid on for the remainder. I think Jim's estimate for 20 minutes total is probably about right (it is good to develop a sense for how the meat should feel when it is done). If you have a good meat thermometer, take the meat off the grill when it hits about 135-140 degrees, depending on your taste Let them rest for about 5 minutes after you remove from the heat, before you cut into them. It will rise another 5-10 degrees while resting.