I'm putting my large recipe website into Wordpress. I am putting everything in a separate folder sitename/wordpress/etc. Will I run into a problem when I'm finished, and it's time to take everything and put it into the root folder?
If you currently have static html pages and want to change your site to use WordPress you can reference external pages from within WP, but you will not gain anything from that setup. To make your pages searchable from within your WP site they would need to be changed to pages or posts manually. I changed a static html site to WP about a year ago and it really took off, but it was a small site with about 35 pages and took a month to convert page by page. It was not the only thing I was doing at the time or it would have been done sooner, but it does take time to manually convert pages, especially if they are not already in xhtml format using utf-8. WordPress can be installed in your site's root directory but it can also be installed in its own directory on an existing domain, just a little more adjustment for existing links that way. Basically it all depends on how you want to use WP and what you want the finished structure of your site to be. If you are not familiar with WordPress I recommend setting up a free account at wordpress.com to learn more about how it works and what it can do for you. Their free sites are only for non-commercial personal sites, but whatever you create there can be exported for integration with your own install when you're ready.
Will I run into a problem when I'm finished, and it's time to take everything and put it into the root folder?
When you move it out to root, in your settings in Wordpress you'll have to change the URL's there. This will immediately log you out of admin (because the URL is now different.) Otherwise, if you're using seo friendly permalinks, no, everything will change from example.com/wordpress to example.com. Also if you're using seo friendly permalinks you may have to modify your .htaccess but most of the time it's relative to the current directory.
That is, if your .htaccess points to /wordpress/index.php, you'd change it to /index.php. If it's just index.php (relative to wherever you are) no change should be needed.
If you're on windows using a rewrite module, it's the web.config file but the concepts are the same.
When you go live with it be sure to set the privacy settings for "allow S.E.'s to index my site" (you have that off, now, during development . . . right?) :-)
Well, it's generally a minor thing and you wouldn't think a directory that has no links to it can be found. But sometimes they are, I've seen it happen and still don't know why. Maybe it's some of the internal workings of Wordpress - pingbacks, etc., and they get linked to elsewhere.