I do not remove any leaves from my aloes unless/until they are dry and brown and let go of the stem easily. I don't think it helps the plant and it may actually hurt the plant.
The question about how the leaves got mushy is a good one. Was this related to rainfall, maybe? Is there any way you can post a picture?
How long have you had the plant? I would think that fertilizer would be fine. You need to measure carefully and not apply too much. I would recommend 1/4 teaspoon in 1 gallon water for regular use or 1 teaspoon in 1 gallon for occasional use. If you do fertilize it's very important to flush the soil when you water, so that water comes out the bottom of the pot, in order for the salt to not build up in the container.
A comment I wrote for that plant in the database here, in case it is helpful:
https://garden.org/thread/post...
