RSX was the 7sp group, when RX100 STI levers were introduced it was a full 8sp groupset, can't speak to rx100 downtube levers as that's older then my involvement in fixing bikes.
The rear derailleur wasn't really designed for it but I'm running a 28t with the same generation shimano 600 and it doesn't mind that much, just make sure you get the chain right, that 53/28 gets dangerous if you don't change the chain at the same time and size it to handle the largest combination. If you want a better range you can change the rear der to an acera or alivio 8 or 9 speed version and run a 12-32. Sram makes a pg830 with an 11-26 and shimano makes an hg50 with an 11-28.