Thanks, I can use your answer later, but you missed the exact point of my question. I do not know that the user will delete specifically the X Location channel. Instead, I want to present a list of channels from all selected objects to the user, so that he will select, for example, "I want to delete the Z Location from all selected objects". However, I could not find a way to convert location[2] to "Z Location".
Re-read the question appears to be a pretty good attempt at an answer to part i. Generally the idea is to have one question per question , re part 2
Can get the name of the property via the "RNA" eg if an object has an fcurve with datapath "location" or "rotation_euler"
>>> C.object.bl_rna.properties['location'].name
'Location'
>>> C.object.bl_rna.properties['rotation_euler'].name
'Euler Rotation'
>>> C.object.bl_rna.properties['rotation_euler'].subtype
'EULER'
>>> C.object.bl_rna.properties['location'].subtype
'TRANSLATION'
As for the axis name from index, have asked this question Python RNA getting the names of Vector axes. (eg "g" is color[1])
>>> index = 0
>>> "XYZ"[index]
'X'
and as for your third question, with espanol selected as language.
>>> bpy.app.translations.pgettext("Location")
'Posición'
consult the docs. https://docs.blender.org/api/current/bpy.app.translations.html#module-bpy.app.translations