new dog and home alone problems

We got a new dog. She is around 5 years old. She was in shelter before and in different homes until she got adopted.
We are teaching her to walk on leash and different commands. She has problems with fear (motor noises, vacuum cleaner...)
Problem i don't know how to deal is when she is left alone.
We left her in room and she peed and destroyed a little bit of door and wall. We don't really have idea how to teach her to be calm when we are gone.
We don't say goodbye to her when we leave, we wait until she is calm to pet her when we come home (we don't pet her immediately). We are teaching her to not follow us everywhere in home. She is getting used to her pet bed. She also whines when i(or someone else who is alone with her) go to another room but she calms herself and goes to her bed soon. I am slowly teaching her to stay in her bed while i'm going to another room.

I found this article teach dog to be home alone and i was wondering if this could help??

We can leave her in living room with cat if that could help her? What else could we do? I read some things about this problem but i don't know how or where to start. I really need a plan (what to do, when to do, how many times- something like this article where it is told what exactly we need to do)