The red pin? It's a verified address if i'm not mistaken? the red rectangle I've never seen before? Only in photoshopped edits.

i did that red box so not to confuse my question with sitelink categories above lol...

you see the the categories in red box?:
Term Dates - ‎School Meals - ‎School Brochure - ‎School Info

how to do that?

As far as I understand it - that's from using a sitemap in xml format - just search for "XML Sitemap" in the Wordpress plugin repository.

Well, I guess I also want to know about whats going on over here because when I search my page over google something like this appears too..

I believe this is something achievable via rich snippets. Have you heard about the strategy? if no, kindly google it, it is something that can help you to tag ratings, reviews etc about your business and this is displayed in the SERPs itself. It can help you to create a call to action.

Hi,

The schema tags are behind this . you can search about it here Product - schema.org to understand it.

My site has this sometimes, for some search terms... I didnt do anything to get it and one of the links is to one of my least popular pages. It seems to be copying my top navigation bar.

That's Google Sitelinks in the red box.

The BIG list of links below the domain is Mega-Sitelinks.

Has nothing to do with schema, it's all internal links + relevancy connecting authority pages.