The widget errors are the only ‘real’ errors. The header related errors are just caused by the error notices being output before the HTTP headers (including the one output in misc). Fixing that error would solve the others.
The problem with the widgets is that they are written with out of date PHP (it doesn’t appear to have been updated in over 4 years). You can read more about that here: http://php.net/manual/en/migration70.deprecated.php
Un-registering the widgets is probably the simplest solution. You can do that in a child theme with the unregister_widget() function:
unregister_widget( 'Thematic_Widget_Search' );
If you need those specific widgets, then you’ll need to unregister them, then re-register them with your own code for them that doesn’t have the constructor issue.
Ultimately though the issue is that the theme isn’t PHP7 compatible, and isn’t maintained any longer. You’ll probably want to find a different theme.