Try watering more, until water comes out the holes. It should not take a lot of water given the depth of the container.
Stop moving the plant from one place to another if it's falling over, and stop replanting the aloe regularly (that will really stress the roots). Either location sounds like it will have sufficient light. A couple of medium sized rocks on either side will help with stability and will not cause any problems for the roots. If you've seen the roots get longer, that's a very good sign, but try to leave them alone in the future.
I do not ever cut off aloe leaves until they are completely dead and brown. I don't think there's any advantage to doing so. I don't know if the brown spots will spread but my guess is they will not, if you provide strong light and water (in moderation) and stability. The more roots your plant has, the better it will be at reinflating the leaves, the more stable it will be in the pot, and the less likely it will be to have any further complications.