Your soil should feel dry at the surface within a day or two. If it's taking weeks, then you need to repot with a more porous soil. You probably have root rot which is why it's not growing. The wet soil could also be rotting the lower leaves that are resting on it. I would remove all of the old soil from the plant, trim away any rotten roots, and repot using a gritty soil. I did this to an Aloe Vera that had the same problem a couple months ago and it hasn't grown yet, but hasn't looked any worse either. Recovery may be very slow during the winter with few roots, but with time it should come back better than ever. Try to give it lots of light since that can be a problem indoors. Here's a link to a very good soil recipe. I've found the specifics are not that important as long as the particles are all large enough that water runs right through them. If you don't have the time or patience to find the ingredients, at least find one gritty component and mix 1-2 parts of it with each part new potting soil. Here is a link that might be useful: Succulent soil mix

The first is Aloe vera v. chinensis. (blooms orange) The 2nd (although that looks like more spots than usual) which is much larger, thicker leaves & yellow blooms, is Aloe vera/barbadensis. I believe it has been reclassified with the latter name, but most people are still calling it by the former (vera), so one or the other is listed as a synonym. See link below for a pic of an Aloe vera/barbadensis ~Rosemarie, who hangs out on the C&S forum, but pops over here on occasion =) Here is a link that might be useful:

There are hundreds of Aloe species, and many more cultivars and unnamed hybrids. There can also be some variability within a species. Pictures would help, but even then IDs may be elusive. Most Aloe veras I know of have yellow, or sometimes orange flowers. The patch growing near at the old property may have been another species, such as arborescens.

Congrats on the new plant and welcome to the forums! 1. Looks physical damage to me. Maybe it was slightly damaged in transport and the leaf eventually buckled? 2. You can keep the leaf as is. It may still grow and stay alive for quite a long time. Eventually the damage portion will dry out and the leaf will break off at that point. Unfortunately, the leaf will never 'heal' or 'regrow' from that point as it was before. On the bright side, over time it will be replaced as your Aloe plant grows larger. Side note: If the terracotta pot has a drain hole, you can eventually remove the plastic pot and plant your Aloe directly in it. That would better if you plan on keeping your plant inside (or outside depending where you life).