As you are using an active directory or local directory, you should be able to search for *arriaga* for example. It may take a while until you see the user details.
Make sure you have the active directory set up in QMC > System > Setup > Directory Service Connectors > Active Directory (or Local Directory).
For the documents, it does not matter if you don't have a Publisher license as long as you have an Enterprise Edition license, that allows you to use DMS instead of NTLM (see QMC > System > Setup > QlikView Server > Security > Authorization). If you have it set up to NTFS, you will not be able to add or remove users, as the permissions are handled by the operating system, that is, you have instead to go to the QVW, right click on it, Properties > Security > Edit.
If you have DMS, go to QMC > Documents > User Documents > Authorization > Named Users > click on the people icon > use *arriaga* and click on the magnifier icon (not just "enter") and click on Add > OK and then Apply, otherwise the changes will not be kept.
And with respect to your subquestions:
"Why does the User Management list appears empty before I search anything?" - Imagine the QlikView server being connected to multiple directories with tens of thousands of AD users (yes, they do exist). Do you want the QMC to get filled with all user details, groups membership details, license assignments, task and document permissions for all users before you can even tell the service that you're only interested in one? I don't think you would like to wait for the listing to be completed every time you open the Users view. I guess it's a matter of not wasting an administrators time (in addition to providing well-targeted tools).
"Why can only I search in this section by the user structure (name.lastname) and not by name (image 1)?" - I would like that too, but I guess that in order to improve search quality (there's usually only one correct answer) and speed (account search is quicker than an all-text-and-description search), searches are restricted to the real unique account names in the various directory types. Moreover, given that all directory types offer different types of details in differently named fields (sometimes even no additional details at all), it's about the only available characteristic that you can be sure about.
Note that as Miguel already stated, you can still use wildcard characters in your search string in case you do not really know the exact account name.