Wooden pieces are all there and in pretty good condx. but some
improper fasteners (which look like modified plaster-board screws have
rusted away). Am wondering if it would be good idea to also use glue
where wood meets wood?

I use Titebond III exterior (Green label) and screws for outdoor
woodwork
http://www.titebond.com/WNTitebondIIITB.asp