You access your member connect mail from the opening page of Ancestry. In the top right hand corner there is a green square, a green leaf, username, upgrade, and "get help".
When you click on the green square a drop down menu comes up with recent contacts. Go to the bottom of the pop up and click "See all Messages". A new window will open with both incoming and out going email. There are basic folders on the left Inbox Sent, Delete and crate new folder.
I have created folders for each of my families.
As said above you can request a copy of your mail when you compose a new message.
From the main page you can search for contacts, by clicking Collaborate. From the drop down memu, select "Member Directory" Because the member name appears on all trees, I use the tab "Basic Information" as I can put the member name in the search window and find the tree owner.