I started to look around "searchbox" properties and found out that in both my Joomla template and Fireboard template the same name for id #searchbox{} was used. As soon as the same id could not be used more than one time on a page, searchbox in Fireboard had the same properties as my Joomla searchbox. As soon as I changed this id in Joomla template and CSS the problem was fixed.
Updated my template for Joomla to fix this problem.
Now I'll start changing template for forum to match Joomla color set.
New template for Joomla is attached to this post.
By the way, Firefox Web Developer toolbar is excellent addon and helps to evaluate and fix problems. Especially CSS section, where you can disable some stiles or view
CSS properties for any part of HTML page.