Hello Juan,
You can check the following:
Typing into the searchbox depends on how big the standard SAP menu is in your system - when starting to type something in to the quicklaunch bar, a searching is started automatically in the background also in the SAP menu structure - you can check what kind of searches are happening if you check what kind of search providers are checked.
For the other parts, can you give a closer example?
Best regards
Melinda