Soft cello cases or gig bags have a pocket on the front of the instrument for the bow - kerioboe is absolutely right, it pays to take the bow out first before unpacking the cello, then put the cello in the case after you have finished using it BEFORE you put the bow away........otherwise a broken bow is a very real possibility (can damage the cello too, if it digs into the cello!). Hard cases are similar to violin cases - they have a couple of clips to hold the bow so that it is kept separate from the cello. Haven't seen a duster in a cello case, though I have seen cloths or covers in violin cases - not sure why this is, other than violins being fragile little flowers?
Rosin - there is usually a pocket on the front of a soft case to put rosin and mutes and such like (soft cloth to clean cello etc). My soft case also has a zip compartment on the back for music and spare strings. Hard case had a section with a flap cover inside the case where you could stow rosin and the rest (new hard case is on my wish list as old hard case is ENORMOUSLY heavy and new cello doesn't fit in it.....now there's a trap for young players, not all full size cellos are the same size

Not that it would have stopped me buying the new cello - and anyway, we can't fit both daughter's 3/4 and my full size in the car boot if either was in a hard case).
Cellist, this is a reasonable question (especially for one who is just starting out, as you are) - I wouldn't want my bow in with my cello without some sort of protection (as you get when you use a case, as stated above) as rosin off the hair isn't great for the varnish or the strings and the bow could easily scratch the cello or be damaged by the cello. Hope this helps and is what you were after!

How are you going with beating the family and friends off your cello? Have they realised yet that you are deadly serious now and touching the cello could result in very unpleasant consequences?

Hope you are enjoying your cello journey.
Wolfnotes