makarakri for the changes you're trying to accomplish, you will probably need to use the Flarum javascript extension API.
You can fork core to modify the javascript Mithril components, but that's a lot of work to maintain. The best is to extend the components from a custom Flarum extension. If the change is important, you can override the entire view method of a component, but ideally you'll use the extend helper so you can modify the HTML in virtual dom of a component, or inject a new component in the virtual dom.
If you are referring to the discussion list, the components you are looking for are probably around DiscussionList and DiscussionListItem. If you are referring to the area where the tags are displayed at the top of an individual discussion, you are probably looking for DiscussionHero.
You could take a look at how the Tags extension adds its tag labels, you'll probably need very similar code if you want to add a button next to it. These areas are built using ItemLists, a system to easily insert items before or after another.
There's certainly a learning curve to the Flarum frontend extensibility. The basics can be found here https://docs.flarum.org/extend/frontend
