Ahrefs is a worthy tool, although I feel they should give new users a free trial instead of the $7 for 7 days trial, which is hardly enough trial period to adequately track a competitor’s website. For individuals/freelancers in the SEO digital space, the subscription pricing is quite high when you’re just starting, but once your clients start to see the benefits, it will be worth it as you get more business. Otherwise, there’s little I found to dislike about Ahrefs.
Basically, Google shows the autocomplete suggestions whenever you start typing anything into Google search box. It is in Google's best interest to show the most relevant keywords in the autocomplete suggestions. Keywords that would help Google to retrieve the most relevant websites and help users find the most relevant content for their search query.

What would be the purpose of/reason for moving back to a different url? If its been a few years, I’d leave it alone unless you watched everything decline since moving to the main url. Moving the forum to a new url now would probably be a bit chaotic, not only for your main url but for the forum itself…. Only reason I could imagine myself moving the forum in this scenario would be if all those links were really awful and unrelated to the url it currently sits on…