Updating Links with Pictures On Homepage

Hello first I want to say I have just started really recently and this is all new to me. I have an idea for my blog but not sure how to go about it and I seem to not be able to find the exact answer to my question.

Part 1:
Is there a way to have on the homepage pictures with headings that when clicked either expand or open a post for a specific category?

Part 2:
If so would it be possible that if I publish a new post in that category that the corresponding categories link’s image changes with the post’s heading?

Sorry if it’s confusing but I cant figure out exactly how to explain it. Any help would be great, especially in layman terms.