Can you please break this out into TWO things. I don't know what your confusion about ALERTS is as most of what you are asking about has nothing to do with alert functionality.
Alerts for showcase are no different than alerts for anything in XenForo. When you are alerted, they show up in the Core XenForo "Alerts". The Alerts in the dropdown only stay in there for a very short period of time after you've viewed the Alerts Dropdown. The stay visible longer on the physical alerts page. Viewing the content does not make an alert disappear.
I "Approve" the post and then Update Moderation queue
I don't understand this at all. How can you update something and then update it again in the moderation queue? If you approve an item OUTSIDE of the queue, then it gets REMOVED from the queue. Sounds like you have a BROWSER Tab open to the Moderation queue, then you manually approve an item via inline moderation or on the item page itself, then go to the tab and do the same thing? Thats the only way I can think of that you could do BOTH at the same time.
When a Showcase Item is in the Mod Queue and the Showcase Item has an associated discussion thread, BOTH of those will be in the Moderation Queue. You ONLY have to approve the Showcase Item. Approving the SC Item will at the same time, handle the associated discussion thread.
As for the Category List. There is some CACHING going on to reduce DB queries, so you can't rely on low level data like that being real time. Also, there are dozens of other things that could come into play depending on your Configuration settings. You could (for example) have settings to NOT include Items in sub categories from displaying which would only display the parent content, but the Parent Content Count on the Cat Nav includes all children. Another would be setting the option to display content with images only and you have content without images. Another would be including the SLIDER and the setting to not include content in the listing that is already being displayed in the Slider. So as you can see, its not always going to be a perfect MATCH (its not designed to be). Its really only noticeable when first starting out. Once you have dozens of items in your categories, you won't even notice.