I have created three content types and associated with a list. When you click the list and then the add new button from the Ribbon in the top, all 3 types display. However, when you click add new item (bottom link on list), there is no drop-down to change content type from within the new form. I have searched through all options and cannot even see a column called content type. I can see Content Type in the views, but no option to add it to the new form. Does anyone have any insight on this issue?
This is the default behavior. The "Add new item" link uses the default content type and does not display a dropdown list to change that value.
Rather than including the Content Type dropdown in the NewForm.aspx, instead you could edit the AllItems.aspx page to include your additional links at the bottom of your page. You'll need to do this in SharePoint Designer and note that you'll need to pass in the content type id in the url to the form.
Be sure that content type matches your list type. So list type is document library then your content type should be document library type (you select the type when you create content type) and same goes for list type (list then item in the ct)