Do we have any standard for using ghost buttons? I know basic details on when to use ghost buttons but more detailed explanation will help Can we use ghost button as well as solid button on same page?
If you notice, there is close icon next to close button. Do you think its very smalll and it should also be button? That close is to close page.
Ghost buttons are primarily a visual treatment of an element. In this regard there isn't a direct reason not to use it for a drop-down menu. Compared to solid buttons ghost buttons have low visual prominence. In that regard, ghost buttons can be used in conjunction with solid buttons if it supports the brand, existing UI and especially the visual hierarchy of the element in question.
The X does not need to be a button, as it acts as a signifier of the natural behavior of a drop-down, which is that a click or tap outside of it will close it.
Drop the x button completely.
Otherwise it is a matter of consistent UI controls across the site/app. If all other UI controls are solid consider making this solid as well, if they are 'ghost' styled then stick with that.
They don't all have to be styled the same but should follow a discernible pattern. The pattern should be obvious to someone who has never used it before (even if your users are all dedicated and trained on your tool).
