Well it depends why are you making web site or for what purpose. The official url should not contain any other url address behind its back. you should have to continue from beginning by purchasing domain name.

Would it even be understood as "your" website? Wouldn't it just be treated as a new part of the existing site, so they get all the benefit?

i think you are talking about getting backlinks from a popular website and yes that will benefit your site in SEO if you have good do-follow backlinks from other site

n all likelihood, you're having a placebo effect. It's applied false logic. You've tried to cheat with multiple IP addresses, and you've noticed your ranks rising. However, it's unlikely that this was the catalyst for your new rankings. The content of the site, quality of inbound links, and other factors are the more probable real reasons for improved rankings. In some cases, it's not that you've gained rank, but rather that somebody else (or several somebodies) have lost theirs. Their site(s) could be getting stale/abandoned, or maybe it was devalued for being caught for implementing black-hat strategies.