Of course that's what Paul Schneider intended for Romulus and Remus. Why wouldn't he? It's hardly a rare practice for English-speakers to use their own name for a foreign land or location rather than the indigenous name, e.g. India instead of Bharat or Mount McKinley instead of Denali. And astronomers have been naming stars and planets after figures from classical mythology for millennia, so it's natural that Schneider assumed they would continue to do so in the future.
Besides, "Balance of Terror" stated clearly that humans and Romulans had very little contact. They never met face-to-face, so it stands to reason that they didn't have any significant cultural or anthropological exchange. In that context, it's entirely plausible that humans didn't even know the Romulans' name for themselves or their worlds.
As for Vulcan, that was originally used by Urbain Le Verrier in the 19th century as the proposed name for a hypothetical planet closer to the Sun than Mercury, proposed to explain irregularities in Mercury's orbit but eventually debunked when General Relativity explained those irregularities instead. However, the planet continued to be featured in science fiction well after it was discredited in science. Given that Roddenberry's original proposal was that Spock was "probably half-Martian," he was evidently thinking of a Solar-planet origin for Spock when he started out. So I kind of suspect the original intent was for "Vulcan" to be Le Verrier's Vulcan, before it was settled that it should be an extrasolar planet.
Also, I'm not sure it was Roddenberry who came up with the name "Vulcan," seeing that it wasn't established until "Mudd's Women" by Stephen Kandel.