How to add custom submit logic to search_block_form?
You can do this in hook_search_preprocess():


  Preprocess text for search.
  
  This hook is called to preprocess both the text added to the search
  index and the keywords users have submitted for searching. The same
  processing needs to be applied to both so that searches will find
  matches.


This is not part of the preprocess in theming. This hook is called directly from the search module. Btw. German umlauts are already replaced by default.