No, you are not allowed to travel between Schengen countries. However, you are very unlikely to get caught if travelling by plane, since no border checks take place. Within Schengen, except for Ryanair, most airlines only check that you're the person on the ticket, and furthermore, if travelling on a full-service airline such as Brussels airlines or Alitalia and you check in online and only travel with hand luggage, chances are you will never have to show your passport at all.
As for travelling by car or especially by bus, I strongly recommend you not to. Buses are likely to travel via Switzerland, which does border checks fairly often at major land crossings such as at St-Louis (where the bus is likely to cross). Once they see you don't have the documents to enter Switzerland, you'll be thrown off the bus, and since they share a building with the French border police at St-Louis, you would likely garner the attention to the French. You could then be in big trouble.
If you choose to travel by car, avoid Switzerland, for the same reason.
Like I said, you are not allowed to travel to other Schengen countries without the actual residence permit card, but in practice it is possible, especially by air on a full-service airline (or at least not Ryanair)
An approval letter for a residence permit is useless by the way, you need the actual card. A letter can be retained even if the approval is revoked, and as such proves nothing. While in the case of a border check the authorities should contact the Belgian authorities to clear things up, in my experience they do not normally bother to, instead summarily denying the person entry.