Good questions, and the only answer that I can give is: it depends.
Given the size of your company though, I'm a little surprised you don't have a dedicated person (contractor or otherwise) working on the web site.
In my experience, it seems that companies who have "information only" web sites, can get by with far fewer resources (web programmers / admins / etc) than companies who derive some sort of income from their site. If you're running an eCommerce site, or other revenue generating scheme, you need someone to 1) keep your site up to date, and 2) make sure the site is running. Down time = money in those cases. You may not need someone "dedicated" full time to the site, and usually with smaller businesses, they have a marketing / web programmer position who keeps a close eye on the web site, but also does other things.
For a web site that is purely informational (which it sounds like yours is), changing content usually gets assigned to a staff person who knows dreamweaver / front page and maybe some basic HTML. If a brochure doesn't get posted on Monday, it's "usually" not critical (unless you have a mass mailing going out that references something that needs to be on the site, but you know what I'm getting at.).
As far as where the position goes in the structure of an organization, that's all over the board too, and I've found usually depends on what senior management thinks a web site is. If they think its "techy", its usually under the IT / Finance deparment. If they think it's more a marketing tool, it goes to Marketing. More often than not though, I find them to be in IT / Finance.
As far as when to make it a full time position, I'd say that if there's more than 75% of a FT equivilant workload for web development, then you need to create the position. IE, if you can say that you're consistantly working 30 hours / week on web only stuff, then you need a web developer position.