Design views display and views filters separately?
Views results (sometimes in Fields Format), Exposed Filters, Attachments are all output into their own HTML containers. But you can override each View output in templates (look under Theme: Information under Advanced View settings). 

For whatever purpose you can create two different Views, or two Displays in the same View, define Fields in one, and define Filters in the other. Then later you can export both, combine and replace the relevant parts of the export into one and Import with an override of the existing View so you end up with a combined configuration. 

But why would you want to separate them in the first place? You can add exposed Filters to a View, work on CSS to style them, then later add Fields and style them. 

It isn't necessary to have the exact same Fields a in View that you want to use as Filters. Fields and Filters are independent. Maybe this is what has been tripping you up? 

You can have Filters that filter out one Content type and also filter by creation date. While in Fields show Titles, Author and Taxonomy terms. A completely different set of information in those sections. 

A lot of confusion may be coming from nomenclature. Views Displays are the different outputs you create according to your needs (Page Display, Block Display). Fields are bits of information defined in Content types. Views Results can be displayed as Fields. Filters control qualitatively what is shown in Views, if you expose them then users can change that themselves (Exposed Filters you usually see above Results that let you control the filters on the fly).