Also in case of the German "Umlaute" you should be able to simply type a/o/u instead of ä/ö/ü in most searches. Searching for "Jurgen" would for example also yield all "Jürgen" in the database.
Afaik, a coherent phonetic search does not exist for CiviCRM yet.
You can use MySQL wildcards in most search fields. Using a percentage sign will find either character - e.g. G%nther will find Gunther, Günther, and Guenther. Note that this slows down searches, especially if it's at the very front of the search.
It looks like force=1 url searching works for profiles too, so just provide a menu link to the page with force=1 on the end of the url, e.g. https://your.site/civicrm/profile?gid=1&reset=1&force=1
Can she create an API to enter a phone number and output a name
If I understand the question right, the API is already available. If you simply need an api that provides the name based on phone number, you can use -
$contact = civicrm_api3('Phone', 'get', [
'sequential' => 1,
'return' => ["contact_id"],
'is_primary' => 1,