In the first issue, if you close the app, the icon will remain there even if you sign out and sign back in. The same happens with all apps. I suggest that when you close the app in that red X icon, it removes the icon in our names as well and it disappears for everyone in the chat.
While using another account, I noticed that when you open the app or when you're playing with someone, the icon doesn't show to the other users. It just shows to the person you're playing with if they open the app as well. It should show to everyone if you have the app open.
In the second issue, after closing the app of the game you opened in the Open apps menu, the icon will remain there and if you click on it, it will show the old version of that app. If you have the current version opened, if you click on the icon without closing the app it will open the old version of the app as well.
When closing an app it could force the icon to disappear, without having to sign out and in.
Bonus: The smileybar disappears sometimes and saying (:$) without parentheses doesn't turn to redface, that probably happens with the other smileys. https://prnt.sc/qfbuf7
* Step 1: Open an app.
* Step 2: Close this app.
* Step 3: Sign out and sign back in.
* Step 1: Close the app you previously opened in the Open apps menu.
* Step 2: Open it again by clicking in the icon you have in your name.
Second issue: https://i.imgur.com/1Tx6qeg.mp4