The 1st link in the article should link to the most important page on site? Does that work?

When publishing articles do you try to keep the 1st link in the article as a link to the most important page on your site?

It depends on the page. If it looks natural than yes.

You can do, if it makes sense(has the same intent) to link to that page. The value depends on the page and also prefer other types of links.

I try to only interlink naturally.
If a link is from another website then yes.

Better to a highly relevant article on your site

I can't see that being a thing..

From what i've seen, the link juice is spread evenly betwen multiple links.

The only time when link priority takes place is when you have 2 links going to the same page, google will accept the 1st link as the real one, and kind of ignore the 2nd one. E.G if you have 2 links, one called "bats" and one called "cars" both pointing to the same page, google will think that page it's linking too has something to do with bats since that is the 1st link on the page.

Yeah, almost any relevant article.

Use unique content on your site. And to add the interlink in your site.

you should do interlinking naturally to the most relevant post

Ask 10 SEOs that question: 4 will say yes, 4 will say no and the other two have never thought about it. Chances are none of the 10 have actually tested it.

I can't seem to find a link but there was a case study I saw recently where someone tested this principle and swapped their internal link structure around with the assumption the first links carried more weight. Rankings increased however, it was a single site so there could have been external factors or they might have just improved their internal linking regardless of the actual position.

I've heard the theory but I don't think there's a solid case study one way or the other. I am for my first link to be at a page I want to rank but I manage this maybe 60-70% of the time. If I can't get it to fit in the content early on then I won't worry about it too much.

As per my experience Yes!
I think the first link is an important and positive impact on that page but I suggest please try to add your first link in the middle paragraph of the content. In most cases, If you are adding a link in the first paragraph then maybe google considers that link as a promotional link.