I would not recommend using the search console removal tool or the robots.txt file. The former is rather tedious and inefficient, I mean how many times you have to do that? while the later is not recommended, if another different page links to the category page your category page will be indexed.
Google knows pretty well when a content is referred from another page. They recommend using excerpts or a summary of your article in the homepage or category pages. Watch this video where Matt from Google explained it.
The best and more efficient way to manage the content of your category pages is to place the following meta robots tag in the head section:
<meta name=”robots” content=”noindex,follow”>
<meta name=”robots” content=”noindex”>
This way Google will no index the page but will follow the links and index the individual posts. Read more about it here.
In regards to the homepage you don’t need to worry. In fact, it is good for the homepage to have content that sometimes gets updated.
Unless you are searching for that specific page, Google will assign the page rank accordingly and will determine the relevant authority to the original content. It will also understand that the homepage is referring or linking to an inner page.