Both the operations you are wanting use AJAX scripts. When a page is displayed WordPress assigns what it calls a nonce (a unique key) to each piece of code that uses an AJAX call. The problem with nonces is that they have a limited life. If the page is left on the screen for a while, the nonce can time out leading to it being denied access to the task and ending in the message you see.
So that should explain the original situation,
Now moving on – I have just tried deleting a custom icon and it all seems to be working fine so there is a strong chance that you are also experiencing some sort of script conflict causing the operation not to complete. Can you try it again but this time with the web browser console open looking for any reported errors.
If you need help with that then this FAQ tries to explain how to do it – it is very simple!
https://simple-press.com/documentation/codex/faq/how-to/how-to-use-your-web-browser-console/