In reply to ianghogg:
General Info:
You should never really use the extreme opposite gear ratios, especially small front, small back. It increases chain wear and the chance of the chain coming off. With that in mind a lot of mechanics set up a bike so that it will rub very slightly at the extremes to get a better shift.
Usual causes of rubbing are your cable is too loose or tight or your limit screws (the two little screws you mention) are set wrong. if the cable is set wrong, you can usually adjust it a little bit via a "barrel adjuster" at the shifter on the handlebars or at the headtube.
Specific to your case:
If your downtube shifters are non-indexed (they don't click into place) and you can't stop the chain from rubbing then the derailleur could be twisted or bent, or maybe at the wrong height on the seat tube. Keep in mind that on a non-indexed derailleur the "shift point" may well be different from the "no rub" point. Though it sounds like you have done a lot of playing around with it.
Can you see on the derailleur cage where the chain rubs?
There is also a (small) chance that you have the wrong size of chain or derailleur.
You never have to live with a rubbing chain! Please don't!