between "sobre" and "encima" I think that Spanish behaves like English. You can say "My dog sleeps on the bed" or "My dog sleeps on top of the bed". Same meaning. And Yes., in Spanish, we can also say "Mi perro duerme EN la cama", which can also mean "on the bed" ... however Spanish then introduces ambiguity if you use "en" since "en" can also mean "in" or "at"... As in any language, if a chosen word is ambiguous, the speak would choose a way to remove that ambiguity.
There is no English "me" or Spanish 'me' in this exercise.
The subject (person or thing performing the action) is "dog"/'perro'
The verb form 'duerme' does not contain a pronoun. The verb used here is 'dormir' which is a stem-changing or irregular verb.
There is a different reflexive verb 'dorimerse'
You have typed a sentence into the forum that has a match in the list of Duo's Correct Solutions for this exercise, and without seeing your screen I can't offer any reason why Duo rejected it.
If possible, please capture a screenshot to share with the user forum so that we can try to see what went wrong.
[GUIDE] How To create, upload, share a screenshot (using postimages.org)
For now, at least be assured that your translation here in the forum is good.
Perhaps it was the 'type what you hear'-type of exercise? Or does that one's 'discuss' button lead to a separate topic?
I know I get bitten by those a lot; leaving me stumped for more than a few seconds after the 'wrong!'-horn before realizing it wasn't asking for a translation.