If you add an LTM and have discovery option enabled, it will discover and delete virtual servers on LTM. I usually don't use that since we dont have wide-ip for all VS's on LTM.
I did not understand the second question. From my understanding with GTM sync group if the server is down, it is marked down on all GTM's as long they are in contact with each other. Only one of them is responsible for monitoring i.e. based on proximity. It could be any an LTM too. It is auto selected. This is my current setup and working as expected.
Nimbostratus