1. In SharePoint 2010 we used Metadata columns, Metadata navigation and column filtering at the top of the list/library to filter the data down by a few columns but now in SharePoint 2013 that doesn't seem to work. You can narrow your results using Metadata navigation but then when you choose to filter on a column the page still shows all of the items instead of narrowing down to only those with the filter value.  
2. Our lists and libraries all have some core columns that use the same lists for lookup values.  We want to filter multiple libraries on one or more of the columns and display the results on one page.
3. The filter web part is great for one library no big deal that you need to add it three times for three different column filters.  The problem is that it requires typing in the strings for the column values exactly without the luxury of a drop down list.  And if you want to do the same on multiple lists/libs then the filter web part needs to be added three times for each if you want to allow a search on three different columns. 
4. Is there an out of box way to enter the filter criteria once but have the results of each library be shown in the page below without custom development.  For example:

Search for Name __________________  and Search for City ___________________ Click button or hit Enter

List 1 Results

Library 1 Results

Library 2 Results

Hi

From your description, my understanding is that:

1. After you enable Metadata navigation in a library, column filtering at the top of the list/library is not working.

I have done a test in my SharePoint 2013 and I can't reproduce the issue.

I enabled Metadata navigation in a library, column filtering at the top of the list/library is still working.

Whether all the column filtering are not working. Check what type of column's filtering is not working. I suggest you create a new list/library and compare the results.

2. I am afraid that we can't filter multiple libraries on one or more of the columns and display the results on one page.

3. You can use SharePoint List Filter web part to with a drop down list of a column to filter a list/library.

4. You can use Refinement web part to make each library items showing in a search results web part.

If I misunderstanding your issue, please offer a screenshot about you issue.

Hope that helpful.

Best Regards

Lisa

My question about out of box query on multiple libraries and then display results on one page has been answered.  Thank you.

The filtering with metadata navigation is still a strange problem and I have seen it reported here:   https://social.technet.microsoft.com/Forums/sharepoint/en-US/78881468-308d-4305-8454-c67fceb05fea/managed-metadata-navigation-tree-in-document-libraries-does-not-allow-further-filtering-on-other   

Maybe we have too many customizations going on with this server.  This is happening on both old and new sites both upgraded from 2010 to 2013 and brand new 2013 team sites:

1. I have two choice columns selected to be in the metadata navigation on the quick launch. 
2. I am filtering via metadata navigation on one choice column and I am trying to use the column filter at the top of the library to filter on the other choice column. 
3. When I select the value to filter on in the column above the library, page supposed to refresh the results.
4. Once you check the box next to the value but the little dialog box never goes away unless you hit the close button. 
5. The system shows the filter icon on both columns but you can see that all items are still showing in the second column the filter has never been applied to the results.

Hi

I have done a test again in my SharePoint 2013 and I reproduced your issue.

When I add column to Key Filters,  the column's filter at the top of the library is missing:

When I add the column to Navigation Hierarchies, then filter to column from the column navigation, the column's filter at the top of the library is missing 

Best Regards

Lisa Chen