i think there are flashers as well as different units you can install to correct the speedo, but it's just an annoyance

for an integra you should be looking for something with a speed rating of at least H-- top speed 130mph, though due to real world everyday pressures i'd look for something with a V rating

as for the tire sizes, what width is the rim, i'm guessing it's 6.5 or 7 inch, which would require as dave said a 205/45-R16, the service number you want should be 86+ with a H,V or possibly W rating, Y tires are too much for an integra