I keep aloe vera near the kitchen stove. If I get a burn -and think about it- I snip off the leaf near the base and just squeeze it. I don't think I'd try to keep any of the gel in advance of actual use. I actually find cold tea bags better for the burn. The growth rate will vary due to the usual reasons: variety of plant; availability of appropriate nourishment and water; sufficient sunlight; size of pot; etc. Having said all that, I divided my 24" tall plant last fall and just stuck one of the babies into a little pot that better fit onto the window sill. It was less than an inch tall at the time, with about a half-inch of root. It is now about 8 inches tall, has 8 leaves, and has made several babies. Apparently it is a happy plant, but I'm going to have to sit down and discuss birth control with it.

Well, a bigger pot holds more soil and the soil holds the water. That can lead to rot, and you don't want that. By mixing perlite with your cactus soil, you can improve the odds of the plant not sitting in water. Mix 1 part soil to 1 part perlite and you'll have it cut 50%. I have my Aloe vera in an 8 inch orchid pot. I'm sure it was overpotted at first. But I have used a good draining mix for the plant and it stays outside all year. The top dressing of rocks helped hold it up until the roots took hold well. You can see the outer leaves are starting to yellow and die back, but the middle growth is still good.

No, leaf starts will not work for the genus Aloe. You need an entire cross section of stem to root. Propagate by removing offsets (pups). There are two forms of Aloe vera. One makes orange flowers, the other yellow. The latter is sometimes called Aloe barbadensis, and is thought to be the original form of the species before domestication. It is relatively rare.

It looks like the outer-most leaf, the oldest leaf of the plant. It will be the first to atrophy and be absorbed by the plant as new growth occurs. It looks quite normal to me, although I can see the mechanical damage you have described. I would leave it alone, at least for now. However, if you really find it to be unsightly and it bothers you, you can cut it off. And perhaps use the Aloe gel for another purpose.