Can we add Short Text Entry field inside in front of normal dropdown?

Dropdown  [           ]            Short Text Entry

To my understanding, you would like to have a textbox as a drop-down field option so respondents will have the ability to add their own input. We do not have a feature where you can add a textbox field as a drop-down option. Only the Multiple Choice fields (checkbox) and Single Choice fields (radio button) have an "Other" option.

May I suggest you also check the User Contributed Dropdown widget and see if it meets your requirements?