How do I get 2 separate displays of a view in one page that are displaying contents from two different content types?
If You are not comfortable with Panels, then


Create one of the required views, a view page (for Content-type A) with proper contextual filters.
Create the other view (Content-type B) as an attachment (attachment Views).

Select INHERIT contextual filters while creating it.
Basically, both the views should have the same contextual filters.

Attach it to the first view created for Content-type A.


This might be an alternate solution.

For overriding the default taxonomy term paths check this tutorial -
Overriding the default taxonomy page with a view in Drupal 7


Hope you are able to manipulate the contextual filters properly in the views.