"Is it still possible to change a tree name? I changed a tree name years ago but now I cannot find that process listed?"
Yes
On any page of the specific tree (I include this if you maintain more than one tree in your account) you would like to change the name of:
Pull down menu *Tree Pages*
Hit the link *Tree Settings* in the pull down menu *Tree Pages*, there you can change the tree name.
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"I would also like to ask that when people communicate that they give the name of the tree, the persons name and a birthdate so that we can quickly locate who they are asking about. "
I include this same request on MY user profile page. It's worked to some degree of success.
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"I have also wondered why when you look at some profiles you can actually click on their tree to go there but most you cannot."
AGHatchett3rd explains (above) where the setting for this feature/function is. It's user choice whether the public tree(s) are listed on their user profile page. The system default is to show them. If they're not listed on the user profile page, the user has specifically chosen to not have their trees listed on their user profile page.