I suggest you add the news article https://schema.org/NewsArticle schema on each of your articles (it will help google understand especialy the "Date" part of the news.
Also you'll have to help google understand those are the last news article on the homepage with some Hn specification.
The current Hn use on your homepage doesn't reflect it. It's a ...
Have you tried it with the aggregate rating Markup in JSON-LD? It looks like that:
Google supports the use of ItemList for their carousel feature. But that does not include Product lists at the moment:
I don't know of any documented features they have for category pages. So no documented rich snippets you can get for those pages.
Your example looks mostly right if you decide to ...