Would "quid nomen tibi est?" be possible? Why exactly is not the verb at the end?
Yes, it's possible because what the important thing it's you use a question intonation. You also could say "Quid est nomen tibi?" and this is the most common way to say it.
You should read this entry regarding the use of "quod nomen" against "quid nomen". https://www.textkit.com/greek-latin-forum/viewtopic.php?t=62864
"Quid est nomen tibi?" is something I remember learning from Wheelock's Latin ages ago.
As it's a common sentence, it's almost an expression, so what would the most common to say it. Let's imagine Latin is not a dead language, how should we say "What's your name" to a Roman girl to sound natural?
I don't hear an aspiration in the q in quid, but I hear an aspiration in the t in tibi!
In general, aspiration doesn't really matter in Latin, unless you're talking about the combinations pt, th, or ch.
What do you mean by "doesn't matter"? If it's not a big matter to discuss or if it was just random in old times for Latin speakers to whether produce aspiration or not?
"What is X to you?" is asking the personal connotations and values you associate with something. Although it can also literally mean "How do you personally define X?" and shades of meaning in between. It depends on what X is.
For example, imagine a child's drawing. Now imagine you're showing it to someone and you ask "What is this to you?" Possible responses include:
"It is nothing to me." (Meaning they have never seen it before and for all they care you can throw it away.)
"I did that when I was five years old."
"My daughter made that when she was four." (If the person gently takes the paper from you and looks at it with a soft smile, this thing has sentimental value to them. They are probably going to keep it somewhere safe.)
It can also literally mean "Tell me your definition of X." For example, "What is love to you?" might be answered with "Love is accepting a person as they are and valuing their happiness."
So "What is a name to you?" is something like "What value do you put on a name? How much do names matter to you? How do you use names in everyday life?" It's somewhat practical, somewhat philosophical.
Would "Quid es tibi" correct, if the noun "namen" is implied (said in the previous sentence for instance)?
Why is the dative case of the personal pronoun used here? Is this a fixed expression?
More or less. Another way to see it is "What name do you have?" since esse + a dative pronoun can mean a person has something. Tibi est frater means You have a brother.
the dative here is used as a dative of possession :)
latin does this sometimes*: you could see a sentence like "the book is for/to you" (liber tibi est) instead of "the book is yours" (liber tuus est). a sentence like that with a dative of possession essentially means "you have the book"
*I'm not exactly sure how common the dative of possession actually is in latin :/ or if it's more common than just using a genitive. but I've seen it quite often :)
If you're using the familiar "tu" (tibi) form, wouldn't you already know their name?
'Tu' is not familiar in Latin, merely singular; by the time T/V distinctions had developed, the people using them were no longer speaking Latin.
What of yours is the name IS a rare consteuction. Is there a fixed pattern for a serie of cases here? Thanks