John Mc,
... (b) both messages a "claimed by" pin on them. What does that
tag mean?
It means that a moderator has edited the message, or has it open for editing now. It is a mechanism intended to help moderators avoid stepping on each other's toes.
(a) When I click on them and say approve, they don't leave the
queue ...
Not sure what you mean by "click on them", but you should be able to check the box on their row and approve them - or open them individually and approve them. In either case you'll get a pop-up warning that the message is claimed.
The hard question- this what I see through my web browser 'at work'.
'at home' I don't see those messages in the pending queue - Pending is
empty form the home browser.
I don't have an answer for that, unless one or the other browser was caching old information, or circumstances changed as you moved from work to home (the messages were approved, deleted, or otherwise removed from the pending queue).
Shal