Since you'll never match the tile, you have two choices...tear out what's there and start over or get a vanity that would cover up the hole. It sounds like you have a 'mudded' wall. This is metal lath (very nasty stuff - sharp edges) with a mortar mix applied on top. This is a premium way of doing a wall not done that much today. If you tear that out, you could replace it with cbu (cement board) then tile. TO get the thickenss right, you may need more than one layer. The best way would be to do another mudded wall, but that is not something you want to try for the first time. Well, you could, but it is not that easy.
For tiling help, check out www.johnbridge.com. They specialize in tiling and you may get some other ideas and help.
As to the bolt, is this the one that sticks out of the wall to hold the sink in place? That is typically a hanger bolt that has a wood screw on one end and a machine screw on the other. There may be enough room to move the sink slightly to install new ones. To get the non-brokend one out, take two nuts and tighten them together to provide a 'head' that you can then unscrew that one. Move it over say 1/2" or so with new holes and bolts. If you want to try to remove the old one, you'd need to drill a hole in the end, then use an EasyOut. This is a tool with left-handed, hardened threads in it. When you screw it into the hole, it will grab the bolt and since it is left-handed, will tighen while loosening the right-hand threaded bolt from the wall.