For a slow shift to the smallest cog, I would think that a half turn (or more) of the barrel adjuster would fix that, if it has one. The rubbing problem is probably the high limit screw, as mentioned. When in the small chain ring though, you will (as far as I know) have to trim the front derailer a bit if you're shifting into the smaller cogs.

Edit: Actually, scratch the whole barrel adjuster thing, that's more for indexing. It's probably just the high limit screws on both the rear and front derailers.