How can I get a taxonomy menu into the main menu AND a block?
Taxonomy menu block combines the idea of Taxonomy menu with the workings of Menu block:

This module does not make use of Drupal's menu system. Taxonomy menu first creates a menu which can then be displayed through a block, this module creates the block directly.

Make multiple menus per vocabulary.

"Split" the taxonomy into different menus. 

Make a menu out of a specific branch of your vocabulary only.

Follow the active trail, also for nodes coupled to a term with a term reference field.