Does that mean I can roll back my function.php file now and remove those settings?
Yep. Those edits are meant to be temporary only and now that you're able to get back in, you can remove the new additions.
Also, regarding my situation. What do you think the simplest way to get my site site at: http://www.site.com/wordpress to be pushed live? Right now it's currently loading http://www.site.com(the old site).
This may not be the simplest way, but it's how I'd do it...
1 - Backup all your files and database and put that somewhere safe.
http://codex.wordpress.org/Backing_Up_Your_WordPress_Files
http://codex.wordpress.org/Backing_Up_Your_Database
http://codex.wordpress.org/Restoring_Your_Database_From_Backup
2 - Make a copy of ALL the files and directories in your /wordpress directory. Make sure you get the whole thing and sub-directories.
3 - Put that copy on your web servers root / and make sure that there are no existing files BEFORE you do this. You don't want to break something else.
4 - Visit http://www.site.com/wordpress/wp-admin/options-general.php and set the WordPress address and site URL to http://www.site.com
5 - Visit http://www.site.com (less the /wordpress) and see if that works.
If this breaks, fix the WordPress address and site url to put it back.
If it works, pat yourself on the back.