Redishing the wheel: a rear wheel is asymmetric. An 8 speed cassette is wider than a 7 speed one (oh, assuming you have a cassette rather than a freewheel, see above mentioned webpage!). So, when you mount a new freehub body on the wheel (the thing the cassette slides onto), you have to fiddle with spacers on the axle and move the rim to the centreline again. The asymmetry is the "dish".
1st shop is assuming he can swap the hub body and the frame accepts the wider axle with 130mm. The only special tools you need is a cassette tool and flat (cone) wrenches, the rest is easy. Overhaul the bearings while you're at it.
http://bikeride.com/freehub-body/ (and replace cables). I think if it actually works with your wheel and frame (measure the distance between the rear dropouts, should be 130mm and the wheel should have a cassette freehub rather than a freewheel) it is a better deal than the second one. Still: check those two points before they order the parts! Will they also replace the rear dérailleur? It is not strictly necessary, but I guess the spring in an 10+ year old dérailleur will be worn.
2nd shop shares my concerns either about frame or about freewheel / freehub issue. Also, 80 bucks per shifter seems a bit steep in my opinion, the new 105 is about 160 EUR for the pair of STIs (about 200 USD I guess). Replacing the shifters means you have to unwrap (part) of the handlebar tape and remove the cables, finally remove the shifters. Then install the new shifters, the cables (maybe replace the housing while you're at it) and rewrap the handlebar tape. Then, tune up shifting + brakes. This should take you also not more than maybe two or three hours if you have not much experience with tuning up shifters but requires no special tools aside from those you need to cut cables and housing (and is not difficult compared to taking apart and reassembling STIs). The howto on this site will be very helpful, especially the videos. Also look at Sheldon Brown's site for cable routing issues etc.
Quote:SHould both shifters be done at the same time? How do you tell if a cable is bad?
Well, you wrote that both are not working... but in general: no, if you don't mind the different look you don't have to replace both. If a cable has any frayed, gunked up, kinked or rusty spots or is really old it is time to replace it (and the housing, only a buck per foot or so). You can also try to isolate the problem by removing the cable from most of the housing but leaving it in the shifter. Then, try to shift and see / feel if it actually pulls and releases the cable. Also try to move the dérailleurs by hand (or maybe pull the cable where it is not inside the housing by hand) and see if they move. If both work, problem is in the middle (cable bad, routed wrong through the shifter or somewhere else).
Oh, and truing a wheel is not that difficult, you just need a lot of patience and time, especially for the first one. Since you don't have the latter, don't try it at the moment.