I created a new section and a custom button, however for admins, this is now causing horizontal scrolling on smaller screens. The Content Editing section has two rows of three buttons each and then a final row of one button. I'm wondering if there's a way for me to force that section to only display rows of two buttons each, so that I'd have more horizontal space for the new section and button?
Look for the item "ContentEditing" in Core DB /sitecore/client/Applications/Launchpad/PageSettings/Buttons/ContentEditing the number of icons/applications to be displayed in each row is specified in this item's RowsNumber field.
You can control the number of items in each row for other groups as well, I mean Marketing Applications, Access Management, etc.
For example - If you update the "RowsNumber" field as 2 below, I guess this might be 3 in your instance.
Then your launchpad will be updated as below to display 2 items per row in the Content Editing group.
In Sitecore 8.2, it doesn't contain the "RowsNumber" logic, so navigate to the launchpad application view, .\Website\sitecore\shell\client\Applications\LaunchPad\Layouts\Renderings\LaunchBar.cshtml and the logic might be common for all the groups including "Content Editing". So if you want to customize for any group then you can add a custom field in Sitecore item [I mean in ContentEditing item], get the value, and write your logic [actually this is implemented out-of-box in newer versions of Sitecore]. Or you can statically set the count as 2 for your ContentEditing group alone. Hope this helps.