Please note Flarum is under development and lots of features are still missing.
Your use case is also very specific. Are you using a broadly available single sign on technology ? You can't expect Flarum to connect to your custom application out of the box / with an add on if you don't have such technology in place.
What is your main website made with? A CMS?
That being said, the extension ecosystem is very powerful and we already have quite a few extensions you can use to achieve your goal. You can also create your own integration!
I have some difficulty searching and pasting links here on mobile but here a few starting points for your issues:
- There's a custom header extension
- You need to do that in your main website I believe, not Flarum
- Have a look at the SSO or Flagrow Passeport extension, eventually as a starting point if they are not compatible with your app
- You should fork the English translation and change the texts to fit
