Adding custom html and css to views header and footer
It isn't good practice to intermix HTML and CSS (embed CSS or use inline styles). As a rule CSS is kept separate in files in your theme, and some in modules. 

Views allows you to define CSS classes for certain parts of the output, but you still have to keep the CSS rules and declarations in your theme. 

If you don't have access to your subtheme for some reason or only want to apply tiny changes you can try CSS Editor. It gives you a place where to enter new CSS rules that will apply throughout your theme. 

You still have to edit your Views output to include the custom HTML classes to elements you want to style, you can do that through Global Text area as you already found. 

Many themes already assign a lot of classes to various Views parts, so maybe you already have classes relevant to your design and you can use them in custom CSS without adding new classes in HTML.