I don't think it will heal, but, if you wanted to leave them the way it is, I don't think you're going to hurt the plant. I've heard that, if you want to get rid of those leaves, to cut a nick into them. This will cause the plant to pull the moisture back into the plant, and withering the leaf. The withered leaf can then be removed without damaging the plant. I've never tried this though. If you decide to cut the leaf off, I suggest sprinkling some cinnamon on the cut afterwards, it acts as an anti-fungal agent.

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:

The brown leaves are probably because of sun. For the smallest leaves do you refer to the outmost leaves or innermost leaves? When Aloe vera is short of water, their outer leaves will become thin and curved inward first. There can be 2 reasons for Aloe vera being short of water - either underwatered or overwatered. For underwater, just water it more. If the latter, it probably got rotted roots which makes the water in the pot unavailable to the plant. May I ask you what kind of soil or mix you put it in? If that's gritty mix or alike, it's not likely to be overwatered. In case you just put it in potting soil or even Cactus soil, you got to check if the soil is still wet inside. You can push a skewer down into your pot till its tip is closer to the pot bottom, wait a minute and pull it out to see if there's a lot moisture on the skewer. If it looks very damp or to the touch, you potting mix probably is not very fast draining. Got to make some change then.

Oldest leaves usually dry up and fall off - just like we 'loose' some hair :) But besides a dry leaf, other leaves seems to be soft & floppy, at least that's what is visible in your photos. What kind of soil is it poted in? Is the saucer attached to the pot? How much-how often do you water? How do you determine it is time to water?