BUT... is there a param to ask to search ALL the fields at once for a big general search?

Search for 1 value in all fields? Nope. You can append more fields to the URL however. Example as follows.

?cb_field1=xyz&cb_field2=abc&cb_field3=123

So I suppose if you append all fields with the same search value it'd give the behavior you're wanting, but it probably won't match anything.

And by the way... is there a list of those params somewhere?

Yup, please see CB > Field Management and review the "name" column. Please keep in mind your field must be set to be searchable and depending on userlist configuration it likely has to be displayed on the list it self as well.

With that below, if xyz is not in the field1 there will be no result for field2 right?

No, it's not an OR scenario but an AND scenario. It's looking that field1 and field2 match the value given. The best way to get the parameters you want is search your userlist as normal and just copy the URL in the browser after search.

I would like to search for a keyword in ANY fields, in one simple search (à la google)

You can't, we've no feature for this currently. CB 2.0 will be redoing the userlist experience and likely allow for such scenarios.

Also here, when i say params i don't mean fields but OPTIONS to use in the URL of COM_COMPROFILER

index.php?option=com_comprofiler&task=usersList&listid=LIST_ID_HERE