Hello @vytaulla51
First, let me clear the confusion you are having with the Local SEO Settings and the Rich Snippets.
These both are for the Schema Markups which help the Search Engines to return more informative results for the users. The Schema Markup tells Search Engines what the data on your site means, not just what it says. Kind of a relevancy signal.
So say for example: if someone searches ABC Hairstylist (ABC being the name of the business), then one would want the Search Engines to show the information of their business in the Search Results, like Opening Hours, price, phone number, etc.
And if someone searches XYZ hairstyle by ABC hairstylist, then the Search Engines should show the information of that particular hairstyle by the ABC Hairstylist. Here the business opening hours, phone number, etc. data is not needed.
So, in this case, you can select LocalBusiness or Hair Salon in the Local SEO Settings and in the XYZ hairstyle post/page you can select Service, Article or Review in the Rich Snippet settings of the post. If you are just going to provide the information of the XYZ hairstyle, then you can use Article Schema else if it’s going to include price, ratings, etc. then you can use the Service Schema.
I hope this clears the confusion.
Now, let’s get back to your questions:
1. My client has a “My Business” (formerly Google Local) listing which appears to have the same info as does the “Local Business” tab in Titles & Meta. Do these co-exist or does the Rank Math listing replace the Google My Business listing in SERPS or are they unrelated or something else?
No, Rank Math doesn’t replace Google My Business. In fact, it should be related. So, if in Google My Business you’ve selected Local Business, then please select the same in the Local SEO Settings.
2. The Local Business listing isn’t the same thing as a rich snippet, right?
They both are Schema Markups. The Local SEO Settings tells the Search Engine what your site is about whereas the Rich Snippet Settings tells the Search Engines what that particular page is about.
3. If #2 is correct, I WOULD want to apply a rich snippet to the site homepage, right? Would there be a reason I wouldn’t? My goal is for her entry to stand out when she comes up in the local serps.
4. If #3 is correct, I’d choose “service” as the rich snippet type for the homepage, wouldn’t I? My client is a hair stylist. It seems like my choice would be Service or Article – article being the default.
It depends on what data you are showing on the homepage. I would suggest you to not use any s\Schema from the rich snippet on the site’s homepage for the reason I explained at the beginning of this answer.
5. What is the purpose of the shortcode in the Service Rich Snippet? Why would I put on a page? Must I?
You can use the Shortcode to display the Schema data on the frontend to your readers. Google requires rating data to be visible on the frontend if you’ve selected Rating or Services Schema.
6. In your help article here: https://rankmath.com/kb/rich-snippets/ you show a list of rich snippet types to choose from which includes “local business” but when I go to a post or page in the site and try to choose what type of rich snippet to apply I do not see “local business” but I DO see “service” (and all the others like Article, Book, Recipe, etc.) which you do not describe at all in the help article.
Sorry about that. It’s a mistake. We’ll update the document soon. The LocalBusiness Schema is not needed in the Rich Snippet’s dropdown.
7. Is it necessary to post the link to the Rank Math generated sitemap on the site somewhere (Like in the footer) for it to be found by Google?
No, it’s not needed to post the link of the Sitemap anywhere on the site. You just need to submit it in the Google Search Console. You can also define the path in the Robots.txt file. This is how the Rank Math plugin does it by default:
https://rankmath.com/robots.txt
Hope that helps. If you have any further question(s), please let us know. Thank you.