What speed is the cassette? from the photo it looks to be a 7 speed (hard to tell with the ghosting). if you have a 7 speed cassette, then the freehub may not be forward compatible with 9 speed cassettes.

For the model number, if it is Shimano. this will be stamped on the rear of the parallelogram.

For using brake cable as shifter cables, you can't do this, as the nipple on the brake cable won't fit into the shifter; and they are too large a diameter to fit the gear outers.

For the shifting, and the teeth being worn, although it's hard to tell due to the ghosting, they don't look to bad, ths shifting can be fixed by adjusting, see the Park Tools site http://www.parktool.com/blog/repair-...illeur-systems or Shimano Tech Docs http://techdocs.shimano.com/techdocs/index.jsp for how to.

Would suggest initally just replacing the cables, inner & outer, setting up the RD correctly should fix your issues without needing to buy new parts