What I'd like to do is have a menuitem on my existing website and when you click on it the forum will show in the main area of that web page instead of a whole new page with another layout.
It's all possible. Check our Gallery
section of examples showing embedded miniBB, as well as search our forums
before duplicating your questions please. It's all explained many times already.
MiniBB authors do not provide free advices or recommendations on such custom tasks like yours. It very depends to your own site, files, design etc. and there is no common solution; but YES, it's all possible.