I don’t see the option of making the “posts” page show my latest blog posts, but keeping my home page to a designated page.

wp-admin > Settings > Reading > Front page displays: A static page (select below)

Front page: (set to your designated page)
Posts page: (set to your Posts page)

Additionally, I wanted to customize my blog page to look nicer and change its appearance.

Your theme, Unicon, is a premium theme, not open source or on www.remarpro.com, the documentation they provided you will specify how to customize it, and their support can help better. re: “After purchasing Unicon you’ll get access to our support area, where we get any issues you’re experiencing sorted out.”

The drop down for the “posts page” doesn’t actually list the “posts page” or “blog” as an option.

We had a web developer create the site, so I’m trying to figure things out now. I’ll look into Unicon support, thank you.

The drop down for the “posts page” doesn’t actually list the “posts page” or “blog” as an option.

In wp-admin > Pages create a page called “Blog”, publish it – then revisit the settings and the new “Blog” page will be an option under “Posts page”