Yes, you can apply custom styling to the search bar. Not sure if you have already seen this but if not than I will recommend your to go through Style your App with CSS learning path on Mendix academy.
Also have a look at this forum post for reference.
Hope this helps!
Yes you can.
In your project go to the folder theme /styles/web/sass/app… there you can find the custom sass files (_custom.scss and _custom-variables.scss) where you can make any adjustment you like.
When, like for this example, you want to overwrite a default styling class (like your grid searchbar), you can find it in this folder: theme/styles/web/sass/core/widgets, where you find the sass file ‘_grid.scss’ wich holds the styling of a datagrid, including the search field.
Copy the class you want to overwrite to your _custom.scss file and there you can start styling it.
Don’t forget to compile your sass with Calypso like explained here: Mendix docs