Water pump in question?

I recently purchased a 1997 green and tan Cobra with 120k miles. I am looking for some advise on two problems this otherwise fine car has. It runs HOT, on the L in NORMAL unless I keep the rpm's around 25k then it rides on the M. It has a recent aluminum radiator and I was told the t-stat was replaced. I was also told the water pump was working so temp. gauge was what needed to be replaced. I shot the front water hoses and pipe with a temp.gauge gun and got reading of 190 (pipe in front) to 230 (pipe going over left head) while the temp. gauge was sitting on the right edge of the L. How can I test the water pump with no radiator cap to open and look into to see if water is circulating?
The other problem is the odometer does not work. I replaced just the odometer and it worked for about 200 miles and then stopped. Probably a short but where do I start? Other than that I think I got a great ride for $7,250. It sounds, rides and looks great. The seller also said that because of the long tube headers and the C&L cold air kit the car needed a tune, suggesting that might affect the running temperature.
Thanks in advance for any help:)
 
The way you describe the overheating issue it sounds as if you have a faulty thermostat. I had the same issue a week ago, replaced the thermostat and all is well now. Check the upper radiator hose with the car when it starts getting hot by giving it a squeeze to see if it's pressurized. The temp gauge would not impede the flow of coolant making the temp rise, nor would the cold air or long tubes cause this problem either. If the system does pressurize and the upper hose you can tell is flowing coolant you need to let the car completely cool, then once it's ice cold there is a 1/4" drive piece on the crossover tube that you can remove check to see if the car has been "burped" or if the coolant is low. You should be able to see the flow there also if you start the car when it is cool cause this is a pressurized side. Good luck and keep us posted.