If I go and get a blood test done for nothing specific is it possible the doctor could find I had something?

Absolutely. Happens all the time. Your real question is, "can i find something treatable." the chemical profiler picks up hypercalcemia (1 person in 1000 has years ruined by a parathyroid adenoma), smoldering liver disease (i had family members die unnecessarily of this), and cholesterol problems long before they are symptomatic. There are dozens of other commons. Much of our longevity is due to screening.
Blood test. Yes, we can tell many things from a blood test.
5% false positives. Unrecognized disease may be detected by doing lab tests for no reason, however, you are likely to get false positive result. 5% of the tests generate false positive finding.