If he was seen and diagnosed with an infection, that would explain many of these symptoms, and Bactrim is an excellent antibiotic for these infections.
The other major issue, though, is that these symptoms are concerning that he may need a significant amount of fluid. He may be getting low in fluid, either because of increased fluid loss related to the infection and frequent urination, plus a significant infection will cause a significant increase in need for fluid. If he is low in fluid, that can cause the pounding heart and dizziness.
Checking the blood pressure in the three positions is the best that can be done at home to assess whether he is low in fluid. If you could do it, it would be the change in blood pressure and pulse in the three positions that would be pertinent, rather than the raw numbers. Since you do not have a BP monitor, then the next option would be to try to dramatically increase fluid intake to see if it helps to ease symptoms. So, it would be good to drink at least 2 liters of hydrating fluids, such as sports drinks, to see if it helps ease the pounding chest and dizziness.
There are many other tests that can be done by a doctor to assess hydration status, but it would be necessary to be seen again. If he is able to get improvement in symptoms from hydration, then that is a good sign, but if he does not have improvement in symptoms or if the nausea limits his ability to be drinking this amount of fluid, then he will do better being seen again.
If I can provide any clarification, please let me know.