You once were able to "demote" certain pages from being listed in Google's sitelinks, but Search Console has since removed this feature. Google's algorithm does its best to cater your sitelinks to the searcher's intent. Check out their update here for some more insight.

I would first ask if you are ranking #1 for your brand name. If not, that's where I would start.

If you are, the next question I'd ask is have you submitted a sitemap to Search Console?

If you have submitted a sitemap, the next step would be to make sure your navigation and content structure is obvious, clear, and well-organized. I would also work to internally (and externally) link to the pages you'd most like to see in your sitelinks. Google will more than likely take this as a signal to know which of your pages are the most important. I'm sure there are many other factors that go into which sitelinks display, but those are the few that you have the most control over, in my opinion.

Hope this helps!

At the moment, to my knowledge, Search Console does not necessarily show any local data.

However, like you mentioned, Google My Business has added some awesome data and insights over the past year. My agency has been loving the recent updates that show how many times your listing has appeared, how many users have asked for directions, how many users have called, and more!