There are different kinds of testing.

The current testing that's reported in the media is an antigen tests, that directly tests for the presence of the virus. Once your viral load comes down, you will test negative for the virus. Hence the reason people are declared recovered after having two negative tests in a row.

A test for the presence of antibodies is a serology test. The last time I looked into it (~one week ago), serology tests were under active development, but had not been deployed anywhere in a widespread way.

So, right now the testing will only tell you if you currently have the virus. Coming relatively soon are tests that will tell you if you previously had the virus.

Yes, but I don't think the PCR test is testing for that, currently. It's looking for pieces of viral RNA that are present in your blood. Antibody serology tests aren't readily available anywhere yet. In fact, the CDC is still developing theirs.

Can't edit this, but to be completely correct, it's looking for viral RNA present in samples, not necessarily blood. More commonly sputum or respiratory samples when it comes to coronavirus.

That sort of test is in development but isn't reliable enough for use yet. The tests currently in use detect virus DNA, not antibodies.