Understanding Wordpress Static Home Page?

I'm interested in how my static homepage will appear in the serps.

If you Google "Socius Marketing" you see the following in the serps:

Socius Marketing
sociusmarketing.com/
Based in Tampa, Florida, Socius Marketing is a full-service search engine optimization company that specializes in organic search engine optimization

However, when you click on the link... you see an entirely different "Title Tag" in the browser tab.

I'm currently setting up my static homepage, named "Home". The automatic permalink given is domain.com/home/

I publish the page, go to Wordpress>Settings>Reading and click static homepage, selection "Home"

My questions are as follows:

Question one: How will my company show up in the serps?

Domain
Domain.com/home/
Site Description

or

Domain
Domain.com/
Site Description

No "Home" in the url. I've been told in another thread that Wordpress will override the /home/ and only show domain.com. I know this to be true in the browser bar, but is it true in the serps? and why?

Second Question:

Google "Socius Marketing"

Why is it that the serps title differs from the actual page title tag when clicked on? I'm using Yoast SEO for Wordpress and can customize my title tags. I want my company name to show in the serps as the title tag and not SEO optimized title tag.

I also have access to my day jobs backend of Wordpress. When you google that company, the same as above happens. I can see in Yoast that the SEO optimized title tag is what's written as the serps title, however... only the domain shows in the serps.

Sorry for the long drawn out, confusingly written question.