Filter Entries field by category in the CP?
No, this is currently not possible, but there's a feature request on Craft's feedback site
that you could vote for: "Entry Type" setting for Entries fields.

As Lewis Jenkins mentioned there, you could use his plugin Dynamic Fields as
a workaround solution in the meantime. Something like this should give you a dropdown field with
all "myOtherChannel" entries related to a category "myCategory".

{% set category = craft.categories.group('myCategoryGroup').slug('myCategory') %}
{% set entries = craft.entries.section('myOtherChannel').relatedTo(category).order('title asc') %}

{% for entry in entries %}
    { "value": "{{ entry.id }}", "label": "{{ entry.title|raw }}" }
    {% if not loop.last %}, {% endif %}
{% endfor %}