I have no idea how you could have registered that domain in May when DH already had a registrant on record as of February; that is a question you need to take up with with Dreamhost.
The hosting part of it is a lot easier to understand. If the registrant of record had the DNS for the domain pointed to Dreamhost’s nameservers (which apparently he does/or did), but never established hosting for the domain, then your creating a hosted domain with that name would result in your site being displayed. While this sounds like it might not should happen, it is really very common - I “create hosted domains” all the time for my clients, who own those domains and control the registration records where the DNS is set up, on Dreamhost.
Which site is ultimately shown to a web visitor is dictated by DNS; sites “hosted” under that name can be maintained on multiple servers. The setting-up of a “hosted domain” with the name paulrush.com alone is not what gets it “on the web”; the registrar’s DNS must also point to that hosting.
Unless something is “borked” on the Dreamhost registrar side of things, it looks from the whois records that the registrant of record is the other person and your being able to host that domain at Dreamhost was a coincidental result of that owner’s DNS setting and the fact he had not established hosting at Dreamhost for that domain. This has happened before, though it is fairly rare.
The real question is whether you actually managed to register the domain with Dreamhost in May, though it already had a whois record in February, and that is what you should get sorted with Dreamhost. It doesn’t look as though they treated your “registration” as a “renewal”, as the expiration date doesn’t tend to indicate that.
It looks like you just “hosted” the domain without “registering” it, but I understand from your original post that you did register it - you need to contact DH tech support to get that all straightened out. Good Luck!