Thank you for the good suggestion!
As for important news, we do indeed have a 'News' section that pops up when there's news, e.g. a new release or an important message. We could make this more permanent; I'll mention this to the team.

Regarding notifications on wallet-related actions (like receiving tokens), is this something that you use in another wallet that you could show us for inspiration?

Hi! You'll be pleased to know we added notifications to Chrome browser to alert you of transaction status changes, i.e. pending to completed or failed, displayed as popup messages in Chrome whatever page you're on.