All the items under your “What I want” list can be easily achieved using the Posts List block which is a variation of the Query Loop block, and if you want to, you can create your very own variation – see Extending the Query Loop block in the block editor handbook.
You can check the following tutorial on using the Query Loop block itself or variations of that block such as the Posts List block: https://learn.wordpress.org/tutorial/taking-advantage-of-query-loops/ (try jumping to minutes 5:49).
But specifically regarding adding a category filter and a pagination, here’s how I added them:
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Adding a category filter
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Adding a pagination (at the bottom, or after the posts list)
However, were you aware of the default category archives, e.g. at https://example.com/category/uncategorized/ (or an actual page/URL here at the time of writing), which displays posts from a specific category and includes a pagination to pages 2, 3, etc.?
I asked so because perhaps you could simply customize your existing category template, as shown in this tutorial for a block theme: https://learn.wordpress.org/tutorial/exploring-category-templates-with-block-themes/, without creating a custom archive?