@martijn, you can move the search bar to either left or right by writting some Custom CSS.

Just copy the following snippet of code according to your choice and place it in Appearance -> Customize -> Theme Options -> Advance Options .

#header-extra .search-form {
    float: left; /* to place the search bar on left */
}

#header-extra .search-form {
    float: right; /* to place the search bar on right*/
}

@martijn, this was a quick and easy hack through custom CSS without any extra effort of customization.

Regarding its custom placement in header or footer or in menu this is possible if you can go through customization of the theme. These changes can be customize by making a Child Theme, overriding the parent theme features.

Alternatively, you can consider hiring a customizer to do this custom job for you in a safe way along with required design workarounds needed during customization.

Kindly, let us know if you have any other queries.