On the forum? You can mark an individual section as read by clicking on its icon. You can also mark all messages as read using the link in the footer on all pages or at the bottom of the statistics section on the home page.
No, like private messages. DMs. PMs.
Open them? They'll show as read once you've clicked into them. There's no way to do it in bulk.
Oh well that's kind of annoying
You can deactivate "Your submission has been accepted" PMs from staff, if that's the problem here.
The forum is 3rd-party software so I can't really control how that works. But yes, if you don't want to receive acceptance PMs, you can disable them from your
Edit Profile page on the sites or from the
User CP here on the forum. You will still receive a PM if your submission is rejected regardless of this setting and it does not affect direct member to member PMs. This still won't let you mark them as read in bulk but will reduce the number of PMs you need to open.