IMO, it wouldn't be more natural.
But to clarify the preposition usage in my own head... If she is just sitting on her bed reading something for a little while -- not lying in bed reading or, say, propped up on a pillow to read -- we would use na, otherwise we'd use v? Or is it not that simple? :-/
Thank you! And, for the benefit of others, I will add another thought that just came to mind.
What would be most natural -- at least in the US, it would be standard usage -- is "My cousin reads/is reading in bed." It would be generally be understood as referring to her own bed, but it would also allow for the possibility that she she might read/be reading in whatever bed she happens to be in at the time.
You would. The reflexive pronoun is there because the subject is "receiving the action". She is reading for herself.
If the pronoun was left out, she would be reading for someone else. It would be weird to leave it out when she's in the bed, but if you added for whom she's reading it, it would be possible. For example: Moje sestřenice čte v posteli pohádky své dceři. My cousin reads fairytales in the bed for her daughter.