Yep, Dan you are right. I just flipped the light on out on my balcony and sure enough I saw a HUGE mouse or rat run across it. It ran into my storage unit that is connected to my balcony. I'm too freaked out to open that door and see what kind of damage it's done. Yikes! Not sure how to get rid of it/them but hoping maintenance can! Thanks so much as I was on the lookout for those big wood roaches and even put out boric acid. This would explain why I was still seeing the droppings.

If you place your plant in full sun, you need to do is slowly. As you found out, it will burn. Wean it into the sun just like you would do yourself to get used to the sun. My plants are in full sun all year long, but I do not get as hot as you mentioned. When it is dry my plants turn brownish, but not burnt like you describe. Your plant should do well beside your brightest window, preferably south facing. The sun is generally not as strong after coming through a window, so wean your plant to that area. A gritty, fast draining soil is best. Many use half perlite and half cactus and succulent soil mixed. I am in a very humid area, so a mix of perlite, granite chicken grit and Turface is what I use. Where are you located? Your growing zone and general area will help determine which potting mix may work best for you.

Did you put it out in the sun w/out doing it GRADUALLY? Turning more purple is the plant's reacting to stronger light. Unfortunately, potting it larger was a mistake & makes the plant more vulnerable to root rot. Especially since your mix doesn't seem to have much Perlite. If mine, I'd amend that soil w/ lots more Perlite or Pumice so that it's 50/50 of each parts of each, to be sufficiently fast draining. Is this your first Aloe? Generally speaking, it's recommended that Aloes be potted in a pot size only one or 2 inches bigger than the rootball. My pix below, are different Aloes than A vera. But pls note the proportion of plant to pot size, much smaller than what you're using. (I grow indoors only.) Also, pls note how coarse my mix is. So I'd recommend you amend your soil & then pot it down smaller to a more appropriate sized pot.

If leaf is soggy, especially at the point where it is attached to the main stem, it it overwatered. I am just stating in general, and it would help a lot if you posted a clear, preferably close-up photo of the problem. Few photos would be helpful, showing detail of the leaf where the problem is, and showing whole plant too. Also some more details, like what is the soil plant is in, pot, when and how much you water...