I'm running a small shop using Wordpress and Woocommerce, selling stuff for the smart home and the Mercedes Benz W211 E-Class. Since I also plan to sell software, a forum would be nice. I've tried several Wordpress plugins, none of them being acceptable. So I decided to use flarum, even if that means some integration work - and manual updates, since I'm on a shared hosting without composer support.
schlauHAUS.biz
For the design, I only use custom header, footer, css, and the FoF Links Plugin. While not being perfect, I think the design is close enough to the rest of the website.
For the integration of users authorisation and settings, I use the WP Users Plugin from @AlexanderOMara. It really works great. Basically, every user/customer who is logged in into his customer's account, is logged in into flarum as well and can start writing right away without registering a second account.
I don't want to use Gravatars, so I use a Wordpress Plugin called "Avatar Privacy". It generates geometrical avatars on the server, so no CDN needed, and caches them on the server. I forked the Gravatar-Plugin from AlexanderOMara to create a new Plugin, "flarum-wp-avatar-privacy". It uses those cached files to display the same avatars in Flarum as used in Wordpress. It's my first plugin for flarum.
What's still missing is a Wordpress Comments integration, like @clarkwinkelmann has done in his Wordpress Integration plugin.
The Website is hosted on a shared hoster, called Bitpalast. It is a great hoster, giving you very much for your money, think of it as an insiders tip. But it does not support composer, so each time I want do update or add a plugin I need to download the files and database, do the update locally in xampp, and upload everything again. Cumbersome, but works.