Let me answer the second question first, as that is expected.
Views does not offer any way to modify what terms are included in a taxonomy filter. If you insert a filter for a taxonomy (e.g. the pa_manufacturer taxonomy) then all of the terms of that taxonomy are included in the filter. The only way it would be possible to change that would be by using JavaScript on the front-end to remove the unwanted options.
What you show in your screenshot is not to choose which terms to include in the filter, it is a "backend" Query Filter that specifies which terms to apply to the query for posts every time the View is loaded, it has nothing to do with front-end searches.
Imagine you had a post type that used a "status" custom taxonomy with terms "published" and "unpublished". You could add a Query Filter for the status taxonomy and manually select the "published" term, and then you could display that View wherever you want and it would only ever show "published" posts.
The first question is trickier, because it is not expected, what you report. I'll need to take a closer look at how you set up the Views. You have provided access to your site, so I will do that now, and report back to you soon.
You can see in this screenshot why the results were not updating correctly: the URL parameter that will be added by the front-end filter control needs to correspond to the URL parameter that the Query Filter is listening to.
I updated the product category Query Filter so that it is again set by the filter control (blue), and I have fixed the mismatched parameters for the other taxonomy (red).
The product image is still disappearing though, I'm checking that now.
the problem on the live site occured when i transfered his work and modifications to the live site, what i did was copy the code from the view to the live site