The parktool blog is general in nature and doesn't address some of the questions I asked.
I have since found the installation guide for the Sora ST 3303 shifters and guide for rear derailleur.
For those of you with the same problem, you have the to press the Sora STI button switch in 7 times. The 1.2mm cable with a bullet end is then fed through shifter with large chainwheel correlated to small cog on cassette.
I have to change the cable that controls my rear derailleur, which are STI Sora 8 speed shifters. The RD looks pretty worn too (approx. 10k miles on the bike).
1. Where can I get the "exact cable" that fits this STI 8-speed shifter?
At any LBS. With the exception of Campagnolo, shift cables are pretty much universal. Road shifters and MTB shifters use the same type of cable. Campagnolo shifters need a cable with a slightly smaller molded end, but will work with Shimano shifters, too. Smooth, die-drawn stainless cables are preferred for an extra dollar or two.
2. The originall factory setup is a Sora RD with a long cage matched to a triple chainweel and 8 speed cassette.
(a) Can I upgrade to a Tiagra 4500 GS Triple 9-speed RD?
(b) If so, will it work with my existing 8 speed cassette since it's for a 9-speed?
3. If memory serves, when I feed the new cable through the STI shifter do I
(a) Click the "button shifter" to its lowest setting in correlation with the chain being on the smallest chainweel and smallest cassette cog?
Yes, but the front shifter doesn't matter, unless you're also replacing that shifter.
(b) If so, do I then fasten the cable so there's no play in it?
I'd reposition the barrel adjuster by tightening it all the way, then backing it out one full turn before connecting the new cable. This will give you plenty of range for dialing the cable tension after installation.
(c) Do I then adjust the top limit screw to align the low cog and bottom limit screw to align the high cog?
If you're replacing the RD, then follow the directions at the parktool site. If you're just replacing the cable and leaving your Sora RD on the bike, don't touch the limit screws.
Anything else I need to know to adjust the gears?
At 10k miles, I'd suggest replacing the short loop of housing at the RD. This section gets more worn inside due to the tighter loop, and it also collects more grime due to its location on the bike. The LBS can cut a new section to match in length and can provide you with new ferrules (the turned aluminum ferrules with the squared "flat" end are better). Of course, some people would just as well replace both shifter cables and all housing at this mileage, but this isn't necessarily warranted at this mileage.
desertdork - thank you very much for the detailed reply. The bike rides perfectly and should be good for another 10K!
I did have to press the button switch in order to align the cable hole so I could fish the cable through. I also screwed in the adjuster barrel all the way and backed it out a full turn like you said and it shifts perfectly in all ranges