Do you have search enabled in AdminCP > Settings > Options > Message Searching Options > Search Engine Enabled ?
How about for the usergroups - Usergroup Manager > edit usergroup > Can Search Forums ?
And does it show up on a default style? Create a new style with no parent:
- Styles & Templates > Style Manager > Add New Style
- Parent Style: No Parent Style
- Title: Default vBulletin
- Allow User Selection: Yes
- Save
Then browse the site using that totally default vbulletin style - do you see the search bar now?
We need a test login to see the other problem with the dropdown.