If you get a blood test done for one thing, could that show other things?

Yeah, it's possible. Usually they take blood from pregnant women because pregnant women are likely to have iron deficiency. If your iron levels are low, they'll most likely test for pregnancy too.

No.  Blood tests test for specific things.  Some tests test for only one thing, a check of your ALT (a liver enzyme) for example tests only the ALT levels in your blood. Others test for several related things, a "chem panel" for example will test the level of several specific chemicals in your blood (a "chem 10" at the vet office where I work for example tests glucose, ALT, AST, BUN, Creatinine, BUN/Creatinine ratio, albumin, and globulins).  

But the lab will ONLY run the test that your doctor has ordered.  If your doctor ordered an iron test, the lab will only run the iron test.  They could run a pregnancy test on the same blood sample if the sample were big enough.  But they won't run a pregnancy test unless your doctor specifically orders one.

Many doctors will talk to you about your test results and recommend additional tests based on the results and physical symptoms that they know you have because sometimes test results coupled with physical symptoms or exam findings might cause them to suspect a condition that they weren't originally looking for.  For example, if a routine iron test came back low and you are a sexually active woman of child-bearing age and you had one or more symptoms of pregnancy, your doctor might recommend running a pregnancy test.  But your doctor still has to put in a separate order for that test, and usually they need your consent to do it.