1. Lean means thinned with mineral spirits or turpentine, fat means eg linseed oil. Most painters do a wash with paint diluted with mineral spirits, then paint mixed with linseed oil over that once its dried. I would consider straight paint out of the tube to be the fattest of the fat.
2. Turpentine, mineral spirits and paint thinner all achieve the same thing (solvents). Go with mineral spirits for the price, and less fumes.
3. Brushes: swish around in solvent, squeeze into a rag, then wash with dish soap and a little bit of water in the palm of your hand. I do sign painting with enamels, and I rinse them in mineral spirits, squeeze out, then dunk in transmission fluid so they never dry out. That's when they die. I've heard of storing them in vegetable oil or vaseline as well.
4. Don't put thinners and oils down the drain. Thinner: I use 2 jars. One I keep all my murky thinner in. It will settle out and you can pour off the clean stuff into another jar to use. When you're done painting, swirl it up really well and pour back into your storage jar. You will eventually end up with a big puck of mud in the bottom which you can let dry out and toss in the garbage. If you really need to get rid of solvent or oil, pour it into a paper towel and leave it in a well ventilated area or put it in the garbage.
Hope this helps