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!

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!

Update: Sorry, I didn't pay careful enough attention to DL's condition! No, squarefree doesn't imply condition 3. That's a stronger local condition. But my guess will be that the method Bhargava uses will still give you a positive proportion, though I'm less certain. (Namely: there is still an Euler factor at p given by the total mass of all etale quintic extensions of Q_p satisfying your conditions.)