To be fair, after about 15 years of anime I still often fail at telling wether some characters japanese name is supposed to be their first, or last name.
Names are usually left untranslated, so any hidden meaning that might link them to their character is also not easy to spot.
So at first an anime chars name is pretty much a white piece of paper to me.
Japanese first names seem to suffer the same problem as first names almost all over the world: they are rather generic and fail at their intended purpose... to be a unique identifier for a single person.
Of course there is only a limited number of reasonable combinations that you can use for a name. But when there are 5 people with the same first name in a class (IRL), it will cause problems.
In anime, the artists will make sure there are no two characters with the same name in their show and they have more of a freedom to choose exotic ones too.
However, across different shows a lot of characters with the same first name will show up and it will feel slightly awkward. If all character has is a generic first name, he/she will either blend in with his/her other namesakes over time, or will have to be identified by some other unique feature. In which case the name was useless.
For this reason, I prefer characters with a full first and last name, even though I they are hard to remember.
On the order thing:
When used in an english sentence (such as translation) first name first, last name last... of course.
That way I know which is which no matter what, without having to decide if a name is supposed to be japanese, or western, or whatever notation anyone else may be using.
Otherwise, what will you do when westerners start using japanese sounding names, or the other way around? Or maybe they migrated to another country, but kept their names? How will you tell which notation to use? It may work with the olympics, because every name has a little flag next to it. But that is a luxury you won't encounter too often.
The notation is a part of the language, not of the name. The name only has the attributes 'first' and 'last'.