So, it means literally "Who is your name?"
Like with the name a separate entity...
I wonder if it doesn't take its root in the old animistic religions, that were before islam.
This article (in Indonesian) confirms what you said:
So it explains why it's an error to say "Apa nama kamu?"
Yes you can. But, that is rude in Indonesian culture. Better to use siapa "nama kamu?"
Yes, you can. This is also possible. In spoken form, Indonesian people use "ya?" at the end of the question like this.
"Nama kamu siapa, ya?" The purpose of "ya" is to make it softer and more polite? However, Indonesian use "kamu" to address a person in a case that they have already make a conversation for some times but they forget to introduce themselves properly. Otherwise, it is going to be rude. They will rather use "anda" for formal and first time conversation.
"Ya" simply means Yes. But in this case, the word "Ya" is added as particle of the sentence. It isn't translated. The addition of ya is to make the sentence have a feel of politeness actually. However, not all questions can use ya as thr particle of the sentence.
This is the meaning of particle: Grammar. (in some languages) one of the major form classes, or parts of speech, consisting of words that are neither nouns nor verbs, or of all uninflected words, or the like.
In Bahasa Indonesia, the particles are lah, toh, bah, nian, etc...