All the information I could find highlight using keys when rendering lists, for example:

<ul>
 array.map((item, index) => <li key=index>item</li>)
</ul>

Are there situations other than lists where it's also helpful to provide keys?

Are there downsides to simply providing keys for every non-static element on the page?

You may use keys to reset a component state. See this article for more information : https://reactjs.org/blog/2018/06/07/you-probably-dont-need-derived-state.html#recommendation-fully-uncontrolled-component-with-a-key

A key is recommended for lists in react because react uses this to identify items that are added, changed or removed.

Keys

Keys help React identify which items have changed, are added, or are
removed. Keys should be given to the elements inside the array to give
the elements a stable identity.

An example with a function component:

function MyListComponent (props) 
 const list = props.list;
 const items= list.map((item) =>
 <li key=item.toString()>
 item
 </li>
 );
 return (
 <ul>items</ul>
 );


const array = [1, 2, 3, 4, 5];
ReactDOM.render(
 <MyListComponent list=array />,
 document.getElementById('root')
);

The only rule is that it must be unique among its siblings.

If you don't use a key - if you don't use a key you'll get a warning.

An id (identifier of each item in your list) is best for this purposes - if you don't have any identifier that you can use, an index of each item in the list can be used, but its not recommended because this can cause problems in cases where the order of items change.

Keys only give a performance benefit when used in lists. When one value in a list is changed, React knows to only update that value and not the whole list. Keys are helpful because they help keep track of elements when the index changes (note: only use index as key for last resort because if the index changes (i.e. an element is added to the front/middle of a list) React will have to re render the whole list). You can read more about it here: https://reactjs.org/docs/lists-and-keys.html. As for the downsides, I'm going to take an educated guess and say including an extra key attribute that is not used anywhere will slow down your compilation time an unnoticeable amount