"She asks not to watch the TV while she is sleeping."
Translation:Она просит не смотреть телевизор, пока она спит.
The English sentence sounds bad to me (there has to be some indication of whom she is asking). The Russian sentence is OK though - it means that she asks everyone else not to watch TV while she is sleeping, presumably not to disturb her sleep - e.g. in the case of poor sound insulation.
No. It's either "она просит не смотреть..." or "она просит, чтобы не смотрели...".
In the latter case "чтобы" serves as a "connector" introducing a subordinate clause (whose subject "они"-"they" is implicit). Hence the mandatory comma in this case: all individual clauses must be separated by a comma in Russian.