How to go about a featured post?

Could you explain a little more about the “broken experiences”? Are you referring to the featured post showing up twice – once at the top of page 1 and again in its original position in the queue of posts in the subsequent pages?

Say I decide to show 4 posts on every page and the featured is number 6. It will then show either 3 or 5 posts, breaking a grid type design.

Having it showing up twice, or just get moved in the queue isn’t that big of a deal, as long as the amounts of posts underneath it is the same. So if I have a featured post and 3 posts on page 1, I need it to be able to show 4 regular styled posts on page 2, 3, 4 etc.

I’ve looked into it more closely and the built-in will work just fine. However there is one little tweak I’d like to implement, but don’t know how: The first page shows different amount of posts dependent on whether the featured post is article 1 or not. Can I have it show featured + 3 on first page no matter what?

You should be able to do that with some coding.

You can use the pre_get_posts hook to control the loop but remember to check that you are modifying it at the right spots. So do some checks, for e.g.:

function stham_mod_query( $query ) {
    if ( $query->is_home() && $query->is_main_query() ) {
        // blah blah blah
    }
}
add_action( 'pre_get_posts', 'stham_mod_query' );

You can also use the get_posts() function to get the posts you need by setting some parameters:
WP_Query Pagination Parameters

You would have to limit the number of featured posts displayed to only 1 and ignore all the others since there’s no limit to how many featured posts you can have:
Display Sticky Posts

I’m not a whiz at development but I hope the above helps a little.