I received a 1st Class medical certificate in May 2005. Do I qualify for BasicMed?
Your first-class medical certificate issued in May 2005 granted, since you were under 40 at the time:

First class privileges until May 31, 2006
Second class privileges until May 31, 2006
Third-class privileges until May 31, 2010

Since that last date is after July 14, 2006, then assuming it expired valid and you have not applied for another medical certificate since, you would qualify for BasicMed.