Are you sure the bike and wheels are in good shape? The sliding thing sounds real strange, are you sure the hubs are tight and the wheels are properly atached to the bike?
There is things you can do with the brakes, ANY bike can have good brakes if you know what to do.
The stem. Is ist possible to fit another stem? Ask thor about this, maybe there is another stem you can fit to solve some of the problems.
If you want to buy a new bike I am thinking BF tikit for some reason but the small wheels is not going to make it more comfortable on the cobblestone (it has suspension I think). You could try a sprung saddle to help out with that.
I`d like to see a picture or several of how the bike looks today. Makes the whole thing a lot easyer. It sounds like you are a fairly tall rider?