Responding to your questions...
1) You can replace the default search box with Google's own by editing searchform.php. Just remove the code that's currently there with Google's.
2) To change the font size and footer area's background color you'll need to add some CSS to custom.css located in the theme's CSS folder. What you add depends on what you're trying to change. For example, add this to change the footer's background color:
Note: The footer is considered where the light grey area of the demo where Archive, Categories, Pages, etc. are located.
You'll have to be specific about which text you want to change the font size of before we can give you that code. Generally speaking though it would look something like this:
3. You'll need to check with Breadcrumb NavXT as to what you need to add but the template you'll want to edit is most likely going to be loop-page.php, loop-index.php, loop-archive.php and loop-portfolio.php all of which are located in the theme's includes folder. I imagine you'll want to add the code directly above this snippet:
<a href="<?php echo get_permalink() ?>" rel="bookmark" title="<?php _e('Permanent Link:','avia_framework')?> <?php the_title(); ?>"><?php the_title(); ?></a>
4. You'll need to review Ubermenu's documentation as to what you need to do. Not sure but they may have specific code you need to add to the template in order to get to work or it may be as simple as changing a setting in your Dashboard - again you'll need to find out from them. Broadscope's menu is located in the header.php using this bit of code:
* display the main navigation menu
* modify the output in your wordpress admin backend at appearance->menus
$args = array('fallback_cb' => 'avia_fallback_menu');