I'm not sure what is and isn't written down, but Manjul Bhargava's work on counting quintic fields can be used to get an exact asymptotic for the number of quintic extensions of Q with discriminant a SQUAREFREE integer between 0 and X. In fact, the number of such will be asymptotic to a constant multiple of X. Squarefree discriminant implies your conditions 2 and 3.
This may be massive overkill, for all I know!