The "Add a web part" function is not appearing in my list. I took the proper steps (List tab --> default new webpart) to go to the section but it won't display still. I am the site collection admin as well, I also did it before in the past but I cannot view the web part in my past sites either...
You don't add web parts to a list, you add them to a page. A good thing to keep in mind is that lists and libraries are where you store and manage your data, pages and web parts are for presenting it out.
To add a web part, go to the page you want to add the web part to, then edit the page (you can do this using the Page tab in the ribbon if it's there, or the gear cog to the top right of the page if not).
One recommendation I would make is to be careful about which pages you edit. I generally suggest to people to avoid editing the pages which are created when you add a list or library. These are the pages you see when you go go the list or library in your left hand navigation. In fact, in a SharePoint context I tell people to forget that they are pages at all, just think of them as going to your list or library, and again that is for storing and managing your data, not for presenting it. There are times when you do need to edit these pages, but if you don't know what you're doing it's easy to do more harm than good.
I would recommend instead only editing your site's home page, or adding new pages to it (again using the gear cog to the top right) and editing those. Then you can add/edit/delete all the web parts you want in relative safety.
It sounds like others have had problems with editing pages in SharePoint 2013 and the page just refreshes instead of opening in edit mode and web part properties can't be changed. This is related to Internet Explorer 11 not running in compatibility mode. Read this article and the comment from Waseem Khan in the comments section: http://wendy-neal.com/2014/06/unable-to-modify-sharepoint-2013-web-part-properties/
To run the site in compatibility mode in IE, do the following:
- Press ALT + T when viewing your SharePoint page
- Click Compatibility View Settings in the menu that appears
- Click Add to add the current SharePoint site to your list of compatibility view pages
- Click Close
Also, you can try using Chrome to see if it truly is an IE issue, as opposed to a setting or permissions issue.