Is there a way that we can modify the layout of the search result page.

A normal thing i like to change:

I want to ad columns with widgets to the page. i can't find the page in the Layout Editor.

Script adjustment so i guess this is tricky to ask here:

I want to remove the pull down menu where you can select in what part of the site people want to  search.

I want to remove the icon that shows on the left of every search result.