Say if I wanted to update my homepage, what's the best approach to accomplish this without affecting the user experience on my live site?
Is there a plugin or software that allows me to edit my site locally then have it automatically update my live site? or would I need to resort to a plugin such as duplicator or all-in-one migration?
Since the website is already live I think that the best way to update your homepage is to create a page that you will not add it to the menu. You start customizing and when you are done you chance the homepage to the new page and rename it.
Usually a visitor will browser the webpage by it's menu and the links inside the page and less offten by using the search button so I would guess you are fine with this method. But if in your site people offten use the search ( if you have one) then you should exclude the new page from the search untill finish.