Durin's Ring

[Kuruharan]

But I thought that Moria's doors were closed.

Hmm...remind me to bring that up the next time I get into an argument about this.

However, I expect that like the Black Gates, the Gates of Moria could be opened for those who knew the passwords or had "special business" in the kingdom.

One other question, were the Dwarves of Beleriand living in Moria too?

Yes and no. There were Dwarves who had fled the Blue Mountains in Khazad-dum, however the Blue Mountains had not been totally deserted by their original inhabitants.

Sauron would have to get within the gates somehow.

If Sauron did give the Rings for the Firebeards and the Broadbeams to their leaders in Moria (something I rather doubt) he could have just given them en masse.

However, the point about the gates being "officially" closed does lend weight to the notion that the Elf-smiths distributed the Rings themselves.